Universality in Minor-Closed Graph Classes

Tony Huynh ‡ Bojan Mohar § Robert Šámal ¶ Carsten Thomassen † David R. Wood ‡

September 2, 2021

Abstract

Stanisław Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). János Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain (i) an inﬁnite complete graph as a minor, and (ii) a subdivision of the complete

graph Kt with multiplicity t, for every finite t. On the other hand, we construct a countable graph that contains all countable planar graphs and has several key properties such as linear colouring numbers, linear expansion, and every finite n-vertex subgraph has a balanced separator of size O(√n). The graph is 6 P , where k is the universal treewidth-k countable graph (which we define T ∞ T explicitly), P is the 1-way infinite path, and denotes the strong product. More ∞ generally, for every positive integer t we construct a countable graph that contains every countable Kt-minor-free graph and has the above key properties. Our final contribution is a construction of a countable graph that contains every

countable Kt-minor-free graph as an induced subgraph, has linear colouring numbers and linear expansion, and contains no subdivision of the countably inﬁnite complete graph (implying (ii) above is best possible).

†Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby,

arXiv:2109.00327v1 [math.CO] 1 Sep 2021 Denmark ([email protected]). Research supported by Independent Research Fund Denmark, 8021-002498 AlgoGraph. ‡School of Mathematics, Monash University, Melbourne, Australia ({tony.huynh2,david.wood}@monash.edu). Research supported by the Australian Research Council. §Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada ([email protected]). Research supported in part by the NSERC Discovery Grant R611450 (Canada). ¶Computer Science Institute, Charles University, Prague, Czech Republic ([email protected]). Partially supported by grant 19-21082S of the Czech Science Foundation. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810115). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 823748. MSC Classiﬁcation: 05C10 planar graphs, 05C83 graph minors, 05C63 inﬁnite graphs

1 Contents

1 Introduction3 1.1 Related work ...... 7

2 Preliminaries8 2.1 Deﬁnitions ...... 8 2.2 Extendability ...... 10 2.3 Trees ...... 11 2.4 Treewidth and Simplicial Decompositions ...... 11 2.5 Balanced Separations ...... 15 2.6 Generalised Colouring Numbers ...... 16 2.7 Bounded Expansion ...... 19 2.8 Planar Triangulations ...... 20

3 Forcing a Minor or Subdivision 21 3.1 Limit Lemma ...... 22 3.2 Routing with Jumps ...... 25 3.3 Proof of Theorem 1.1...... 29 3.4 Excluding Complete Graph Subdivisions ...... 31

4 Universality for Trees and Treewidth 32 4.1 Universal Trees ...... 33 4.2 Universality for Treewidth ...... 34 4.3 Graphs Deﬁned by a Tree-Decomposition ...... 37 4.4 Simple Treewidth ...... 38 4.5 Treewidth Extendability ...... 43

5 Treewidth–Path Product Structure 45 5.1 Layered Partitions ...... 45 5.2 Planar Graphs ...... 47 5.3 Graphs on Surfaces ...... 47 5.4 Beyond Minor-Closed Classes ...... 48 5.5 Excluding an Arbitrary Minor ...... 50

6 Avoiding an Inﬁnite Complete Graph Subdivision 51 6.1 Characterisations ...... 51 6.2 Kt-Minor-Free Graphs ...... 52 6.3 Locally Finite Graphs ...... 60 6.4 Forcing Inﬁnite Edge-Connectivity ...... 61

7 Final Remarks 61

2 1 Introduction

A graph1 is planar if it has a drawing in the Euclidean plane (or equivalently on the sphere) with no edge-crossings. Planar graphs are of broad importance in graph theory. The 4-Colour Theorem for colouring maps [13, 141] can be restated as every planar graph is 4-colourable. The 4-Colour Conjecture (as it was) motivated the development of much of 20th century graph theory, and its proof was one of the ﬁrst examples of a computer-based mathematical proof. The Kuratowski–Wagner Theorem [118, 167] characterises planar graphs as those graphs not containing the complete graph K5 or the complete bipartite graph K3,3 as a minor. More generally, the Graph Minor Theorem of Robertson and Seymour [142] says that any proper minor-closed class of graphs can be characterised by a ﬁnite set of excluded minors. Planar graphs are foundational in Robertson and Seymour’s Graph Minor Structure Theorem, which shows that graphs in any proper minor-closed class can be constructed using four types of ingredients, where planar graphs are the most basic building block. Figure 1: The icosahedron graph.

Planar graphs also arise in many areas of mathematics outside of graph theory. The Koebe disc packing theorem [106], which says that every planar graph can be represented by touching discs in the plane, is the beginning of a rich theory of conformal mappings [97, 155]. Steinitz’s Theorem [154] connects planar graphs and classical geometry: the 1-skeletons of polyhedra are exactly the 3-connected planar graphs. Knot diagrams represent knots as planar graphs equipped with over/under relations [40, 117]. Embeddings of graphs on surfaces, especially triangulations, are fundamental objects in hyperbolic geometry [36, 152]. Planar graphs also have applications in topology. For example, Thomassen [161, 162] presented a simple proof of the Jordan Curve Theorem based on planar graphs. Planar graphs also arise in physics. Quantum field theory studies Feynman diagrams, which are graphs describing particle interactions. Planar Feynman diagrams [15, 125] lead to notions such as the ‘planar limit’ of quantum field theories. Indeed, there is a beautiful theory of ‘on-shell diagrams’ which says that scattering amplitudes in N = 4 supersymmetric Yang-Mills theory can be calculated using planar graphs [90]. Planar graphs also arise in computational geometry [85], for example as Delaunay triangulations of point sets, which leads to practical applications such as computer graphics and 3D printing. They are central in graph drawing research [156], which underlies the field of network visualisation.

The focus of this paper is on infinite graphs, which are ubiquitous mathematical objects. For example, they are important in random walks and percolation [86], where it is desirable to avoid finite-size boundary effects. They also arise as group diagrams in combinatorial group theory [123] and low-dimensional topology [83, 153]. Bass–Serre theory analyses the algebraic structure of groups by considering the action of automorphisms on infinite trees [17, 149, 150]. Several group-theoretic results can be proven by considering a group action on infinite graphs. For example, there is a simple proof that every subgroup of a free group is free using infinite graphs and covering spaces [169]. Infinite Cayley graphs are

1Graphs in this paper are simple and either ﬁnite or countably inﬁnite, unless explicitly stated otherwise.

3 central objects in geometric group theory [50]. Infinite planar graphs play a key role in the theory of patterns and tilings [89]. Infinite graphs also arise in nanotechnology [79] and as models of the world wide web [23]. In addition, numerous results for finite graphs have led to extensions for infinite graphs or to interesting open problems. See [53, 109, 130, 159] for surveys on infinite graphs.

Universality for Planar Graphs

This paper addresses universality questions for planar graphs and other more general classes. A graph G contains a graph H if H is isomorphic to some subgraph of G. A graph U is universal for a graph class if U and U contains every graph in . The starting point G ∈ G G for this work is the following question of Ulam: Is there a universal graph for the class of countable planar graphs?

This question was answered in the negative by Pach [134], who proved that no countable planar graph contains every countable planar graph. Pach’s result suggests the following question, which motivates the present paper. Recall that graphs in this paper are countable (that is, ﬁnite or countably inﬁnite), unless explicitly stated otherwise. What is the ‘simplest’ graph that contains every planar graph?

Of course, the answer depends on one’s measure of simplicity. First, it is desirable that a graph that contains every planar graph has bounded chromatic number. By the 4-Colour Theorem [13, 141] and the de Bruijn–Erdős Theorem [49], the complete 4-partite graph2 with each colour class of cardinality contains every planar graph. But this graph is very dense. ℵ0 At the other extreme, one might hope that some graph with bounded genus or excluding some ﬁxed graph as a minor contains every planar graph. Our ﬁrst theorem (proved in Section3) dashes this hope and strengthens the above-mentioned result of Pach [134].

Theorem 1.1. If a (countable) graph U contains every planar graph, then

(a) the inﬁnite complete graph Kℵ0 is a minor of U, and (b) U contains a subdivision of Kt for every t N. ∈

If a graph U contains every planar graph, then Theorem 1.1 says it is impossible for U to exclude a ﬁxed minor or subdivision. However, it is desirable that U satisﬁes many of the key properties of planar graphs. We contend that these properties include the following:

(K1) Every ﬁnite subgraph of U has bounded minimum degree. (K2) Every ﬁnite n-vertex subgraph of U has a balanced separator of order O(√n). (K3) U has bounded r-colouring numbers.

(K4) U contains no subdivision of Kℵ0 . We now justify these key properties. (K1) is the primary indicator of a sparse graph class, and implies several other desirable properties. In particular, if every ﬁnite subgraph of U

2 Let Kn be the complete graph with n vertices. Let Km,n be the complete bipartite graph with m vertices

in one colour class and n vertices in the other colour class. Let Kn1,...,nk be the complete k-partite graph with ni vertices in the i-th colour class.

4 has minimum degree at most d N, then applying a greedy algorithm, every ﬁnite subgraph ∈ of U is (d + 1)-colourable, and in fact is (d + 1)-list-colourable. By the de Bruijn–Erdős Theorem [49], U itself is (d+1)-colourable, and is (d+1)-list-colourable by a similar argument. (K2) implies the Lipton–Tarjan separator theorem [122], which is a seminal result about the structure of planar graphs. (K3) is a robust measure of graph sparsity (deﬁned and discussed in greater depth in Section 2.6). For now, note that (K3) implies (K1), which in turn implies that U has bounded chromatic number. Moreover, (K3) implies that numerous other colouring parameters are bounded in U, including acyclic chromatic number, game chromatic number, etc. More generally, (K3) implies that U has bounded expansion, which is a key property in the Graph Sparsity Theory of Nešetřil and Ossona de Mendez [131]. Finally, in light of Theorem 1.1(b), (K4) is important since planar graphs have no K5 or K3,3 subdivision.

Product Constructions

The second main contribution of this paper is to construct graphs that contain every planar graph and satisfy (K1)–(K3). To describe these graphs we need two ingredients.

The first ingredient is the notion of treewidth, which is a parameter that measures how similar a given graph is to a tree (see Section 2.4 for the definition). For example, a connected graph with at least two vertices has treewidth 1 if and only if it is a tree. Treewidth is of great importance in structural graph theory, especially in Robertson and Seymour’s work on graph minors [142]. The Grid-Minor Theorem of Robertson and Seymour [144] shows that treewidth is related to planarity in the sense that a minor-closed graph class has G bounded treewidth if and only if some finite planar graph is not in . In a sense detailed G below, planar graphs are the smallest minor-closed class with a rich structure. Treewidth and planar graphs are also of great importance in algorithmic graph theory. In particular, many NP-complete problems remain NP-complete for finite planar graphs, but are polynomial-time solvable for classes of finite graphs with bounded treewidth (those excluding some planar graph as a minor). This shows that finite planar graphs often lie at the boundary of hard and easy instances of numerous algorithmic problems.

It is well known that for each k N, there is a graph k that is universal for the class ∈ T of treewidth-k graphs. Section 4.2 gives an explicit deﬁnition of such a graph that is of independent interest and important for the main results that follow.

The second ingredient is the notion of a graph product, as illustrated in Figure2. For graphs G and H, the Cartesian product G H is the graph with vertex-set V (G) V (H), where × vertices (v, w) and (x, y) are adjacent if v = x and wy E(H), or w = y and vx E(G). ∈ ∈ The direct product G H is the graph with vertex-set V (G) V (H), where vertices (v, w) × × and (x, y) are adjacent if vx E(G) and wy E(H). Finally, the strong product G H is ∈ ∈ the union of G H and G H. × With these ingredients in hand, we prove the following, where P∞ is the 1-way inﬁnite path.

Theorem 1.2. Both P∞ and P∞ K contain every planar graph and satisfy T6 T3 3 (K1)–(K3).

5 K P K P K P 4 □ 3 4 × 3 4 ⊠ 3 Figure 2: Cartesian, direct and strong graph products.

We actually prove a strengthening of this result in terms of so-called ‘simple treewidth’. We define a graph that is universal for the class of graphs with simple treewidth k, and we Sk show that can be replaced by in Theorem 1.2. This is noteworthy since is planar but T S S3 is not. These results are proved in Section 5.2. T3 We also establish several results analogous to Theorem 1.2 for more general graph classes. First, consider graphs embeddable on an arbitrary surface. The join A + B is the graph obtained from the disjoint union of graphs A and B by adding all edges uv with u V (A) ∈ and v V (B). We show that for each g N0, each of ( 6 + K1) P∞ Kmax{2g,1}, ∈ ∈ S ( + K ) P∞ K and ( + K ) P∞ contain every graph of Euler genus g, S3 1 max{2g,3} S6 2g and satisfy (K1)–(K3). This result is proved in Section 5.3. We provide similar results for various non-minor-closed classes, such as graphs that can be drawn on a fixed surface with a bounded number of crossings per edge; see Section 5.4. Moreover, for each finite graph X, we construct a graph that contains every X-minor-free graph. This graph is more complicated to describe than the above products, but it still satisfies (K1)–(K3); see Section 5.5.

Graph products also provide a mechanism to construct 4-colourable sparse graphs that contain every planar graph. To see this, note that χ(G H) min χ(G), χ(H) for all × 6 { } graphs G and H. Also note that if X is a k-colourable subgraph of G, then X is isomorphic to a subgraph of G K . Together, these observations imply that if U is any graph that × k contains every planar graph, then U K is 4-colourable and contains every planar graph. × 4 For example, by Theorem 1.2, ( P∞) K is a 4-colourable graph that contains every S6 × 4 planar graph. Moreover, U K inherits many of the desirable properties of U with a small × 4 change in the constants.

Avoiding an Inﬁnite Complete Graph Subdivision

Our final results are related to property (K4), and are presented in Section6. Graphs containing no infinite complete graph subdivision were characterised in Robertson, Seymour, and Thomas [145]. Theorem 1.1(b) says that arbitrarily large (finite) complete graph subdivisions are unavoidable in any graph that contains all planar graphs. On the other hand, we prove that infinite complete graph subdivisions are avoidable. In fact, this result holds for induced subgraphs and in the setting of Kt-minor free graphs.

6 Theorem 1.3. For each t N, there is a graph Ut that contains every Kt-minor-free graph ∈ as an induced subgraph, and satisﬁes (K1), (K3) and (K4).

Since planar graphs contain no K5 minor, the case t = 5 in Theorem 1.3 includes planar graphs. It is open whether Ut satisﬁes (K2), although n-vertex subgraphs of Ut do have balanced separators of size O(n3/4 log n) by a result of Plotkin, Rao, and Smith [138]; see Section 2.7.

1.1 Related work

Before continuing, we brieﬂy survey related work on graph universality, focusing on connec- tions to planar graphs. A graph U is strongly universal for a graph class if U and G ∈ G every graph in is isomorphic to some induced subgraph of U. G Ackermann [3], Erdős and Rényi [71] and Rado [139] independently proved that there exists a graph, now called the Rado graph or the random graph [33–35], that contains every graph as an induced subgraph. Pach [133] went further by showing that there exists a graph U such that every graph G can be isometrically embedded in U; that is, the distance between any two vertices in G equals the distance between the corresponding vertices in U. See [33–35] for surveys on the Rado graph, and see [27, 28, 30, 91, 112, 147] for various related results.

Universal graphs for classes deﬁned by an excluded subgraph have been extensively studied. For example, Henson [98] proved that for each t N there is a strongly universal graph for the ∈ class of Kt-free graphs. Several authors have addressed the question: for which (usually ﬁnite) graphs X, is there is a strongly universal X-free graph [41–45, 80, 81, 92, 108, 110, 111]?

Diestel, Halin, and Vogler [56] considered universality questions for classes excluding Kt as a minor. For t 6 4, they constructed a strongly universal graph, and for t > 5, they showed there is no universal graph. Note that this result (for t > 5) is an immediate consequence of our Theorem 1.1(a). Similarly, Diestel [52] considered universality questions for classes excluding Km,n as a subdivision. For m = 2 and n = 3, Diestel showed there is a universal graph but no strongly universal graph. In the case m = n = 3, the proof of Pach [134] can be extended to show that there is no universal graph. For 2 6 n 6 m and m > 4, Diestel showed that there is no universal graph (thus solving a conjecture of Halin).

Broere, Heidema, and Mihók [29] constructed, for each integer k, a strongly universal graph for the class of graphs with the property that every subgraph is k-degenerate (that is, every finite subgraph has minimum degree at most k). Similarly, for every finite graph H, Mihók, Miškuf, and Semanišin [126] constructed a strongly universal graph for the class of graphs that admit a homomorphism to H. By Euler’s formula, every finite planar graph has minimum degree at most 5, and by the 4-Colour Theorem, every planar graph admits a homomorphism to K4. However, there are 2-degenerate graphs (such as 1-subdivisions of complete graphs) and there are 4-colourable graphs (such as complete 4-partite graphs) that satisfy none of (K2)–(K4). So degeneracy or homomorphisms to a fixed graph H are too broad for our purposes.

Universality questions for ﬁnite graphs are also widely studied (usually for induced subgraphs). Results are known for trees [10, 20, 47, 48, 84], cycles and paths [2], bounded degree graphs

7 [1, 5, 6, 8, 32, 73], graphs with a given number of vertices [4, 11, 21, 129], graphs with a given number of vertices and edges [31, 46], and sparse graphs [16, 19]. The following question has been the focus of attention for ﬁnite planar graphs: what is the least number of vertices in a graph that contains every n-vertex planar graph as an induced subgraph [22, 60, 72, 82, 102]? Improving on a long sequence of results, Dujmović et al. [60] recently obtained an optimal answer for this question, by constructing, for each value of n, a graph with n1+o(1) vertices that contains every n-vertex planar graph as an induced subgraph. Esperet et al. [72] improved this result by constructing a graph with n1+o(1) vertices and edges that contains every n-vertex planar graph as an induced subgraph. Esperet et al. [72] also constructed a graph with (1 + o(1))n vertices and n1+o(1) edges that contains every n-vertex planar graph (as a subgraph). The proofs in [22, 60, 72] all use Theorem 5.6 below, which is also a key tool in our work.

One can also ask universality questions for the minor relation. Robertson, Seymour, and Thomas [146] proved that every n-vertex planar graph is a minor of the planar grid P14nP14n. The question becomes more challenging for infinite planar graphs. For example, Diestel and Kühn [57] observed that the planar graph obtained from K4 by joining to each of its four vertices infinitely many new vertices of degree 1 is not a minor of the infinite grid. Nevertheless, Diestel and Kühn [57] constructed an infinite planar graph U such that every planar graph is a minor of U.

Finally, we mention uncountable graphs. Wagner [168] proved that a graph (of any cardinality) is planar if and only if it has at most continuumly many vertices, no K5 or K3,3 subdivision, and at most countably many vertices of degree at least 3. See [107, 111] for more on universality for uncountable graphs.

2 Preliminaries

This section contains preliminary material, including deﬁnitions (Section 2.1); an introduction to extendability (Section 2.2); universality for trees (Section 2.3); and useful results about simplicial decompositions, chordal graphs and tree-decompositions (Section 2.4). Then we provide more detail about some of the key properties of sparse graph classes that were introduced in Section1. In particular, we look at the relationship between separators and treewidth (Section 2.5), generalised colouring numbers (Section 2.6), and bounded expansion (Section 2.7).

2.1 Deﬁnitions

We use the following notation: N := 1, 2,... and N0 := N 0 . For m, n Z with m n, { } ∪ { } ∈ 6 let [m, n] := m, m + 1, . . . , n and [n] := 1, 2, . . . , n . { } { } We use standard graph-theoretic notation and terminology [55]. V (G) We consider undirected graphs G with vertex set V (G) and edge set E(G) (with ⊆ 2 no loops or parallel edges). A graph G is trivial if E(G) = . A graph G is ﬁnite if V (G) is ∅ ﬁnite, and is countable if V (G) is countable. Recall that, for brevity, we assume all graphs

8 are countable.

The neighbourhood of a vertex v in a graph G is N (v) := w V (G): vw E(G) ; then G { ∈ ∈ } v has degree deg (v) := N (v) . A graph G is k-regular if every vertex of G has degree G | G | k. A graph G is locally ﬁnite if every vertex in G has ﬁnite degree. For S V (G), let ⊆ N (S) := S (N (v) S). G v∈S G \ For a vertex v in a directed graph G, the in-neighbourhood of v is N −(v) := w V (G): G { ∈ wv E(G) and the out-neighbourhood of v is N +(v) := w V (G): vw E(G) . Then v ∈ } G { ∈ ∈ } has in-degree deg−(v) := N −(v) and out-degree deg+(v) := N +(v) . G | G | G | G | An orientation of a graph G is a directed graph obtained from G by directing each edge from one endpoint to the other. An orientation is acyclic if there is no directed cycle. In an acyclically oriented graph, if there is a directed path from a vertex v to a vertex w, then v is an ancestor of w and w is a descendant of v.

A graph H is a subgraph of a graph G if V (H) V (G) and E(H) E(G). Two graphs ⊆ ⊆ G and G are isomorphic if there is a bijection f : V (G ) V (G ) such that for all 1 2 1 → 2 v, w V (G ) we have vw E(G ) if and only if f(v)f(w) E(G ). ∈ 1 ∈ 1 ∈ 2 A set of graphs is a graph class if for every graph G , every graph isomorphic to G is G ∈ G also in . A graph class is monotone if for every G every subgraph of G is in .A G G ∈ G G graph class is hereditary if for every G every induced subgraph of G is in . A graph G ∈ G G class is countable if has countably many equivalence classes under the isomorphism G G relation.

A colouring of a graph G is a function that assigns one ‘colour’ to each vertex of G, such that adjacent vertices of G are assigned distinct colours. For k N, a k-colouring is a colouring ∈ with at most k colours. The chromatic number, χ(G), of G is the minimum k N such that ∈ G is k-colourable. If there is no such k, then G has chromatic number χ(G) = . ∞ A clique in a graph G is a set of pairwise adjacent vertices in G. The clique-number, ω(G), of G is the maximum k N such that G has a clique of cardinality k. If there is no such k, ∈ then G has clique number ω(G) = . ∞ For d N, a graph G is d-degenerate if every ﬁnite subgraph of G has minimum degree at ∈ most d. The degeneracy of G is the minimum d N such that G is d-degenerate. ∈ A vw-path in a graph G is a path with endpoints v and w. We let distG(v, w) be the length of a shortest vw-path in G. If there is no such path, then dist (u, v) = .A vw-path P in G ∞ G is a geodesic if the length of P equals distG(v, w).

The 1-way inﬁnite path P∞ is the graph with vertex-set N and edge set n, n + 1 : n N . {{ } ∈ } The 2-way inﬁnite path C∞ has vertex-set Z and edge set n, n + 1 : n Z . {{ } ∈ } A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. A graph class is minor-closed if for every graph G G , every minor of G is also in . A minor-closed class is proper if some graph is not ∈ G G G in . G A model of a graph H in a graph G is a collection (Xv)v∈V (H) of pairwise disjoint connected subgraphs of G indexed by the vertices of H, such that for each edge vw E(H) there is an ∈

9 edge of G between Xv and Xw. Each subgraph Xv is called a branch set of the model. Note that H is a minor of G if and only if there is a model of H in G.

A subdivision of a graph H is any graph obtained from H by repeatedly applying the following operation: delete an edge vw, introduce a new vertex x, and add new edges vx and xw. A graph G contains an H-subdivision if G contains a subgraph isomorphic to a subdivision of H. In this case, H is a minor of G.

Planar graphs form a proper minor-closed class. More generally, for any surface Σ, the class of graphs embeddable in Σ form a proper minor-closed class. The Euler genus of the orientable surface with h handles is 2h. The Euler genus of the non-orientable surface with c cross-caps is c. The Euler genus of a graph G is the minimum Euler genus of a surface in which G embeds (with no crossings). See [128] for background on embeddings of graphs on surfaces.

If H is a subgraph of a graph G, then a chord of H (with respect to G) is an edge vw E(G) E(H) with v, w V (H). A graph G is chordal if every cycle C of length at ∈ \ ∈ least 4 has a chord.

A vertex-partition, or simply partition, of a graph G is a set of non-empty sets of vertices P in G such that each vertex of G is in exactly one element of . Each element of is called P P a part. A partition of a graph G is ﬁnite if each part of is ﬁnite. The quotient of is P P P the graph, denoted by G/ , with vertex set where distinct parts A, B are adjacent P P ∈ P in G/ if and only if some vertex in A is adjacent in G to some vertex in B. A partition of P G is connected if the subgraph induced by each part is connected. In this case, the quotient is a minor of G (obtained by contracting each part into a single vertex).

A partition of a graph G is chordal if the quotient G/ is chordal. P P A set S of vertices in a graph G is separating if G S is disconnected. Sometimes we also say − G[S] is separating. A separating set S in G is minimal if no proper subset of S is separating. If S is a minimal separating set, then for each component X of G S, each vertex in S has − a neighbour in X.

2.2 Extendability

A graph class Γ is extendable if the following property holds for every graph G: if every ﬁnite subgraph of G is in Γ, then G is in Γ. (Recall that G is assumed to be countable.) The following result of Imrich [100] provides an example of an extendable class.

Lemma 2.1 ([100]). The class of planar graphs is extendable.

Proof. If a finite graph H is a subdivision of a graph G, then H is a subdivision of a finite subgraph of G. That is, if no finite subgraph of G contains a subdivision of H, then H is not a subdivision of G. Kuratowski’s Theorem says that a (countable) graph G is planar if and only if G contains no K5 or K3,3 subdivision. By the above observation, G is planar if and only if no finite subgraph contains a K5 or K3,3 subdivision. That is, G is planar if and only if every finite subgraph of G is planar. Hence planarity is extendable.

10 A function f is a graph parameter if f(G) N for every graph G, and f(G1) = f(G2) for all ∈ isomorphic graphs G1 and G2. A graph parameter is extendable if for every graph G and integer k N, the class G : f(G) k is extendable. ∈ { 6 } The following two lemmas are useful for showing that certain graph parameters are extendable. For example, they each imply that chromatic number is extendable (the de Bruijn–Erdős The- orem [49]).

Lemma 2.2 (König’s Lemma [113]). Let V1,V2,... be an inﬁnite sequence of disjoint non- S empty ﬁnite sets. Let G be a graph with vertex set Vn, such that for all n N every n∈N ∈ vertex in V has a neighbour in V . Then G contains a path v , v ,... with v V for n+1 n 1 2 n ∈ n all n N. ∈ Lemma 2.3 (Zorn’s Lemma; see [113]). If a partially ordered set P has the property that every chain in P has an upper bound in P , then P contains at least one maximal element.

2.3 Trees

A tree is a connected graph with no cycles. An orientation of a tree is a 1-orientation if every vertex has in-degree at most 1. For every directed edge vw in a 1-oriented tree, v is the parent of w and w is a child of v. If there is a directed path from a vertex v to a vertex w in a 1-oriented tree, then v is an ancestor of w and w is a descendant of v. If a vertex r in a 1-oriented tree T has in-degree 0, then every edge of T is oriented away from r, implying that r is the only vertex with in-degree 0. In this case, we say T is rooted at r. In a tree T with a speciﬁed root vertex r, we implicitly consider the edges of T to be oriented away from r. Lemma 2.4. For every vertex r of a tree T there is a 1-orientation of T rooted at r. Moreover, a tree T has an unrooted 1-orientation if and only if T contains a 1-way inﬁnite path.

Proof. For the ﬁrst claim, simply orient every edge of T away from r, to obtain a 1-orientation of T rooted at r.

Now suppose that T contains a 1-way inﬁnite path P starting at some vertex v. Orient every edge of P towards v, and orient every edge of T E(P ) away from P . Then every vertex − has in-degree exactly 1, and T has an unrooted 1-orientation.

Finally, suppose that T has an unrooted 1-orientation. So every vertex has a parent. Let v1 be any vertex in T . Suppose that v1, v2, . . . , vi is an anti-directed path in T (meaning that each edge is oriented from vj to vj−1). Let vi+1 be the parent of vi. So v1, . . . , vi are all descendants of vi+1. Repeating this step, we obtain a 1-way inﬁnite anti-directed path v1, v2,... .

2.4 Treewidth and Simplicial Decompositions

This section introduces graph treewidth and its connection to Halin’s Simplicial Decomposi- tion Theorem via chordal graphs.

11 A tree-decomposition of a graph G is a collection (B V (G): x V (T )) of subsets of V (G) x ⊆ ∈ (called bags) indexed by the nodes of a tree T , such that:

(a) for every edge uv E(G), some bag B contains both u and v, and ∈ x (b) for every vertex v V (G), the set x V (T ): v B induces a non-empty subtree ∈ { ∈ ∈ x} of T .

We emphasise that the subtree in (b) must be connected. If all the bags are finite and their cardinalities are bounded, then the width of a tree-decomposition is the cardinality of the largest bag minus 1. The treewidth of a graph G, denoted by tw(G), is the minimum width of a tree-decomposition of G. These definitions are due to Robertson and Seymour [143]; see [53, 119–121] for work on tree-decompositions of infinite graphs. Treewidth is recognised as the most important measure of how similar a given graph is to a tree. Indeed, a connected graph with at least two vertices has treewidth 1 if and only if it is a tree.

A path-decomposition is a tree-decomposition in which the underlying tree is a path. We denote a path-decomposition by the corresponding sequence of bags (B1,B2,... ). The pathwidth of G, denoted by pw(G), is the minimum width of a path-decomposition of G. The bandwidth of a graph G, denoted by bw(G), is the minimum k N for which there is a ∈ linear ordering v , v ,... of V (G), such that i j k for every edge v v E(G). It is 1 2 | − | 6 i j ∈ easily seen that tw(G) 6 pw(G) 6 bw(G). The general deﬁnition of a simplicial decomposition can be found in [159]. Here, we give

the deﬁnition in the special case of countable graphs containing no Kℵ0 . Let H1,H2,...

be pairwise disjoint (finite or infinite) graphs containing no Kℵ0 and no finite separating complete subgraph. (We consider the empty graph to be complete, so each Hi is connected.) Form a sequence of graphs G1,G2,... as follows. Let G1 := H1. Having defined Gn, define Gn+1 by selecting, for some p N0, a Kp subgraph in Gn and a Kp subgraph in Hn+1, ∈ := S and identifying the two copies of Kp. Let G n∈N Gn. If G contains no Kℵ0 , then H1,H2,... are said to form a simplicial decomposition of G, where each Hi is called a simplicial summand. Halin [93] proved the following (where, as always, we assume graphs are countable):

Theorem 2.5 ([93]). Every graph containing no Kℵ0 has a simplicial decomposition.

The next two lemmas show the equivalence between simplicial decompositions and tree- decompositions, and their relationship with chordal graphs. We include the proofs for completeness.

Lemma 2.6. Let H1,H2,... be pairwise disjoint graphs, none containing an infinite complete subgraph or a finite separating complete subgraph. Then H1,H2,... form a simplicial decomposition of a graph G if and only if there is a tree-decomposition (B : x V (T )) of G x ∈ for some rooted tree T , and a bijection f : V (T ) N such that f(v) < f(w) whenever v is → the parent of w, and G[B ] = H for every x V (T ), and G[B B ] is a finite complete x ∼ f(x) ∈ x ∩ y graph for every xy E(T ). ∈

Proof. Suppose H1,H2,... form a simplicial decomposition of G. We claim that for each n N, the graph Gn deﬁned above has a tree-decomposition (Bx : x V (Tn)) for some ∈ ∈

12 n-vertex tree T rooted at node r, and there is a bijection f : V (T ) [n] such that f(r) = 1 n → and f(v) < f(w) whenever v is the parent of w, and G[B ] = H for every x V (T ). For x ∼ f(x) ∈ n = 1, let T be a 1-node tree T with vertex set r , let f : V (T ) [1] be a bijection, 1 1 { } 1 → and let Bx := V (H1). Consider T1 to be rooted at r. Assume we have the claimed tree- decomposition and bijection for some n N. Then Gn+1 is obtained from Gn by identifying, ∈ for some p N, a Kp subgraph in Gn with a Kp subgraph in Hn+1. The ﬁrst Kp subgraph ∈ is in some bag Bx by Lemma 2.8. Let Tn+1 be the tree obtained form Tn by adding one new node y adjacent to x, where f(y) := n + 1 and By := V (Hn+1). Consider Tn+1 to be rooted + 1 := S at r. We obtain the claimed tree-decomposition and bijection for n . Let T n∈N Tn. Then (B : x V (T )) is the desired tree-decomposition of G and f is the desired bijection. x ∈ Conversely, suppose there is a tree-decomposition (B : x V (T )) of G for some rooted x ∈ tree T , and a bijection f : V (T ) N such that f(v) < f(w) whenever v is the parent → of w, and G[B ] = H for every x V (T ), and G[B B ] is a ﬁnite complete graph x ∼ f(x) ∈ x ∩ y for every xy E(T ). We prove by induction on n N that H1,...,Hn form a simplicial ∈ ∈ decomposition of Gn := G[ x∈V (T ),f(x)∈[n]Bx]. This trivially holds for n = 1. Assume that ∪ −1 H1,...,Hn form a simplicial decomposition of Gn. Let y := f (n + 1). Let x be the parent of y in T . So f(x) < f(y), implying f(x) [n]. By assumption, G[B ] = H . Let ∈ y ∼ n+1 S := Bx By. By assumption, G[S] = Kp for some p N. Thus Gn+1 is obtained from Gn ∩ ∼ ∈ and Hn+1 by identifying S (in Gn) with the image of S under the assumed isomorphism from G[By] to Hn+1. Thus H1,...,Hn,Hn+1 form a simplicial decomposition of Gn+1. And H1,H2,... form a simplicial decomposition of G.

A simplicial orientation of a graph G is an acyclic orientation of G such that the in- neighbourhood of every vertex is a clique.

Lemma 2.7. The following are equivalent for a graph G containing no Kℵ0 : (a) G is chordal; (b) G has a simplicial decomposition in which each simplicial summand is a ﬁnite clique; (c) G has a tree-decomposition in which each bag is a ﬁnite clique; (d) G has a simplicial orientation; (e) every minimal separating set in G is a clique.

Proof. (a) = (b): Assume that G is chordal. By Theorem 2.5, there is a simplicial ⇒ decomposition H1,H2,... of G. Every induced subgraph of a chordal graph is chordal, so each Hi is chordal, implying that every minimal separator in Hi induces a complete subgraph. By assumption, each Hi has no separating complete subgraph. Thus, Hi has no minimal

separator, implying Hi is complete. Finally, Hi is ﬁnite since G has no Kℵ0 . (b) (c) follows immediately from Lemma 2.6. ⇐⇒ (c) = (d): Assume that G has a tree-decomposition (B : x V (T )) in which each bag is ⇒ x ∈ a clique. Root T at an arbitrary node r. For each vertex v V (G), let xv be the node of T ∈0 closest to r such that v Bxv . For each node x V (T ), if Bx := v V (G): xv = x , then ∈ 0 ∈ { ∈ } acyclically orient the clique on Bx. Consider an edge e = vw of G. If xv = xw then we have already oriented e. If xv is closer to r than xw, then orient e from v to w. If xw is closer to

13 r than xv, then orient e from w to v. Now G is acyclically oriented, and for every vertex v V (G), the in-neighbourhood of v is a subset of B and is therefore a clique. ∈ xv (d) = (a): Assume that G has a simplicial orientation. Let C be a cycle in G. So C has ⇒ a vertex v with in-degree 2 in C. The neighbours of v in C are adjacent in G. So C has a chord (unless V (C) = 3). Thus G is chordal. | | (a) = (e): Assume that G is chordal. Let S be a minimal separating set in G. Let X and ⇒ Y be distinct components of G S. Suppose that v and w are non-adjacent vertices in − S. Since S is minimal, there is a shortest vw-path P in G with every internal vertex in X, and there is a shortest vw-path Q in G with every internal vertex in Y . Thus P Q is a ∪ chordless cycle in G. This contradiction shows that S is a clique.

(e) = (a): Assume that every minimal separating set in G is a clique. Let C be a cycle in ⇒ G of length at least 4. Let v, w be non-adjacent vertices in C. Let S be a minimal separating set in G such that v and w are in distinct components of G S. So S is a clique. Each of − the two vw-paths in C have a vertex in S, implying C has a chord. Hence G is chordal.

For a proof of the following elementary and well-known result see [55].

Lemma 2.8. For every graph G containing no Kℵ0 , for every clique X of G, every tree- decomposition of G has a bag containing X.

We emphasise that G must contain no Kℵ0 in Lemma 2.8. For example, if G is the complete graph with V (G) = N, and Bn := [n] for each n N, then (B1,B2,... ) is a ∈ path-decomposition of G in which every bag is ﬁnite, and no bag contains the clique V (G).

A simplicial k-orientation of a chordal graph G is an acyclic orientation of G such that the in-neighbourhood of every vertex is a clique of size at most k. For brevity, a simplicial k- orientation is henceforth called a k-orientation (which matches the deﬁnition of 1-orientation when k = 1). A k-orientation is rooted if each connected component of G has a vertex of in-degree 0. It is easily seen3 that each component of G has at most one vertex of in-degree 0. The next lemma follows from Lemmas 2.7 and 2.8.

Lemma 2.9. The following are equivalent for a graph G and k N: ∈ • G is chordal with no Kk+2 subgraph; • G is chordal and is (k + 1)-colourable; • G has a tree-decomposition in which each bag is a clique of size at most k + 1; • G has a k-orientation.

The next lemma (which is well known when G is ﬁnite) says that every graph has a ‘normalised’ tree-decomposition.

3Say G is a connected chordal graph, oriented so that the in-neighbourhood of each vertex is a clique. Say r is a vertex of in-degree 0. Let v be any vertex of G. Let P be a shortest rv-path in G (ignoring the edge orientation). If there is a vertex x in P incident with two incoming edges yx, zx in P , then yz is an edge, implying that P is not a shortest rv-path. So no vertex in P has two incoming edges in P . The edge incident to r in P is outgoing at r. So P is oriented from r to v. In particular, v has in-degree at least 1. Thus G has at most one vertex of in-degree 0.

14 Lemma 2.10. Let G be a graph with treewidth at most k N. Then there is a tree T with ∈ V (T ) = V (G) and there is a tree-decomposition (B : x V (T )) of G with width at most k x ∈ such that:

• T is rooted at some vertex r and B = 1, | r| • for every edge vw of G, v is an ancestor of w or w is an ancestor of v in T , and • for every non-root node w V (T ), if v is the parent of w in T , then B B = 1. ∈ | w \ v| Moreover, G has a (k + 1)-colouring such that distinct vertices in the same bag are assigned distinct colours.

Proof. Let (B : x V (T )) be an arbitrary tree-decomposition of G with width at most k. x ∈ Let x be an arbitrary node of T . Add a new node r to T only adjacent to x, and let B := v r { } where v is an arbitrary vertex in B . Consider T to be rooted at r. Thus B = 1, as desired. x | r| Consider each non-root node y in T . Let x be the parent node of y. Let t := B B . | y \ x| If t = 0 then contract the edge xy into x. If t 2, then choose one vertex w B B , > ∈ y \ x subdivide the edge xy introducing a new node z in T , and let B := B w . Repeating z y \{ } these operations, we may assume that B B = 1 for every non-root node y of T with | y \ x| parent x. Thus there is a bijection from V (G) to V (T ), where each vertex w of G is mapped to the node x in T closest to r with w B . Rename each vertex of T by the corresponding ∈ x vertex of G. We use r to refer to the vertex of G corresponding to the root of T .

For every edge vw of G, since v and w appear in a common bag, v is an ancestor of w or w is an ancestor of v in T .

We now (k + 1)-colour the vertices w of T in order of their distance from r (breaking ties arbitrarily). First, let c(r) := 1. When we come to colour vertex w, if v is the parent of w then B B = w , implying all the vertices in B w are already coloured. Let w \ v { } w \{ } c(w) be an element of [k + 1] not assigned to any vertex in B w . This is possible since w \{ } B k + 1. We now prove that distinct vertices in the same bag are assigned distinct | w| 6 colours. This is true for the root bag since it only contains one vertex. Consider two vertices v and w in the same bag. Let x be the node of T closest to the root, such that v, w B . ∈ x Then v = x or w = x. Without loss of generality, w = x, implying v B w . By ∈ w \{ } construction, c(w) = c(v). Hence distinct vertices in the same bag are assigned distinct 6 colours.

2.5 Balanced Separations

A separation in a graph G is a pair (G ,G ) of subgraphs of G such that G = G G , 1 2 1 ∪ 2 E(G ) E(G ) = , V (G ) V (G ) = , and V (G ) V (G ) = . The order of (G ,G ) 1 ∩ 2 ∅ 1 \ 2 6 ∅ 2 \ 1 6 ∅ 1 2 is V (G ) V (G ) .A k-separation is a separation of order k. A separation (G ,G ) is | 1 ∩ 2 | 1 2 balanced if V (G ) V (G ) 2 V (G) and V (G ) V (G ) 2 V (G) . | 1 \ 2 | 6 3 | | | 2 \ 1 | 6 3 | | A graph class admits strongly sublinear separators if there exists c R+ and β [0, 1) such G ∈ ∈ that for every graph G , every subgraph H of G has a balanced separation of order at ∈ G most c V (H) β. For example, Lipton and Tarjan [122] proved that the class of planar graphs | | 1 admits strongly sublinear separators (with β = 2 ). More generally, Alon, Seymour, and

15 Thomas [9] proved that every proper minor-closed class admits strongly sublinear separators 1 (again with β = 2 ). Robertson and Seymour [143, (2.6)] established the following connection between treewidth and balanced separations: Every graph G has a balanced separation of order at most tw(G) + 1. Dvořák and Norin [70] proved the following converse: If every ﬁnite subgraph of a graph G has a balanced separation of order at most s, then tw(G) 6 15s. Dvořák, Huynh, Joret, Liu, and Wood [67] proved the following strongly sublinear bound on the treewidth of graph products, which is used in Section5.

Lemma 2.11 ([67]). Let G be an n-vertex subgraph of H P for some graph H and path p P . Then tw(G) 6 2 (tw(H) + 1)n 1, and G has a balanced separation of order at most p − 2 (tw(H) + 1)n.

For k N, a tree-decomposition (Bx : x V (T )) of a graph G has adhesion k if Bx By k ∈ ∈ | ∩ | 6 for every edge xy E(T ). Given a tree-decomposition (B : x V (T )) of a graph G, the ∈ x ∈ torso of x is the graph obtained from G[B ] by adding an edge vw whenever v, w B B x ∈ x ∩ y for some edge xy E(T ) incident to x. If Γ is a class of graphs, then a tree-decomposition ∈ is said to be over Γ if every torso is in Γ. If Γ is a graph, then a tree-decomposition is said to be over Γ if every torso is isomorphic to a subgraph of Γ. For a graph U, let (U) be the D class of graphs that have a tree-decomposition over U. For k N, let k(U) be the class of ∈ D graphs that have a tree-decomposition over U with adhesion k.

The following simple lemma will be used in the proof of Theorem 5.27.

Lemma 2.12. For every graph class Γ and n N, the maximum treewidth of a graph in ∈ (Γ) with at most n vertices equals the maximum treewidth of a graph in Γ with at most n D vertices.

Proof. Let f(n) be the maximum treewidth of a graph in Γ with at most n vertices. It suﬃces to prove that if G is a graph in (Γ) with at most n vertices, then tw(G) f(n). D 6 Let (B : x V (T )) be a tree-decomposition of G such that each torso is in Γ. For each x ∈ node x V (T ), since B n, the torso of x has a tree-decomposition (Cx : z V (T x)) of ∈ | x| 6 z ∈ width at most f(n). Let T ∗ be the tree obtained from the disjoint union of T x taken over all x V (T ), where for each edge xy T , we add an edge between a node α V (T x) and a ∈ ∈ ∈ node β V (T y) such that B B Cx Cy, which exist since B B is a clique in the ∈ x ∩ y ⊆ α ∩ β x ∩ y torso of x and in the torso of y. We obtain a tree-decomposition of G with width at most f(n).

2.6 Generalised Colouring Numbers

Kierstead and Yang [104] introduced the following deﬁnition. For a graph G, total order of V (G), vertex v V (G), and r N, let Sr(G, , v) be the set of vertices w V (G) ∈ ∈ 0 ∈ for which there is a path v = w , w , . . . , w 0 = w of length r [0, r] such that w v and 0 1 r ∈ v wi for all i [r 1]. For a graph G and integer r N, the r-colouring number colr(G) is ≺ ∈ − ∈ the minimum integer such that there is a total order of V (G) with S (G, , v) col (G) | r | 6 r for every vertex v of G.

16 An attractive aspect of generalised colouring numbers is that they interpolate between degeneracy and treewidth [116]. Indeed, it follows from the deﬁnition that col1(G) equals the degeneracy of G plus 1, implying χ(G) 6 col1(G). At the other extreme, Grohe, Kreutzer, Rabinovich, Siebertz, and Stavropoulos [87] showed that colr(G) tw(G) + 1 for all r N, 6 ∈ and indeed lim colr(G) = tw(G) + 1. r→∞

Generalised colouring numbers provide upper bounds on several graph parameters of interest. For example, a graph colouring is acyclic if the union of any two colour classes induces a forest; that is, every cycle is assigned at least three colours. The acyclic chromatic number χa(G) of a graph G is the minimum integer k such that G has an acyclic k-colouring. Acyclic colourings are qualitatively diﬀerent from colourings, since every ﬁnite graph with bounded acyclic chromatic number has bounded average degree. Kierstead and Yang [104] proved that χa(G) 6 col2(G) for every graph G. Other examples include game chromatic number [103, 104], Ramsey numbers [37], oriented chromatic number [114], arrangeability [37], etc.

Generalised colouring numbers are important also because they characterise bounded expansion classes [171], they characterise nowhere dense classes [87], and have several algorithmic applications such as the constant-factor approximation algorithm for domination number by Dvořák [65], and the almost linear-time model-checking algorithm of Grohe, Kreutzer, and Siebertz [88].

A graph class has linear colouring numbers if there is a constant c such that col (G) cr for G r 6 every G and for every r N. For example, van den Heuvel et al. [165] proved the following ∈ G ∈ results (always for every r N): Every finite planar graph G satisfies colr(G) 5r + 1. More ∈ 6 generally, every finite graph G with Euler genus g satisfies colr(G) 6 (4g + 5)r + 2g + 1. Even t−1 more generally, for t > 4, every finite Kt-minor-free graph G satisfies colr(G) 6 2 (2r + 1). The proofs of these results depend on the following lemma, which we will use in our proof of Theorem 6.6.

Lemma 2.13 ([165]). Let G be a ﬁnite graph that has a connected partition = P A ,A ,...,A such that for each part A , there are at most d neighbouring parts { 1 2 n} i ∈ P A N (A ) with j < i, and V (A ) is the union of the vertex-sets of p geodesic paths in j ∈ G/P i i G (A A − ). Then col (G) p(d + 1)(2r + 1). − 1 ∪ · · · ∪ i 1 r 6

We need the following rooted variant of r-colouring number. For a graph G and r N, ∗ ∈ the rooted r-colouring number colr(G) of G is the minimum integer such that for every ordered clique C of G there is a total order of V (G) with C at the start, such that S (G, , v) col∗(G) for every vertex v of G. The next lemma is the motivation for this | r | 6 r deﬁnition.

Lemma 2.14. For every graph U and graph G (U) and r N, ∈ D ∈ ∗ ∗ colr(G) 6 colr(U).

Proof. By assumption, G has a tree-decomposition (B : x V (T )) such that for each node x ∈ x V (T ), the torso G of x is isomorphic to a subgraph of U. Let C be a clique of G. Root ∈ x T at a node s such that C B . Such a node exists by the Helly property of trees. For ⊆ s

17 each vertex v V (G), let h(v) be the node of T closest to s and with v B . Let ∈ ∈ h(v) T be any total order on V (T ), where x y whenever y is a descendant of x. Since G U, ≺T x ⊆ we have col∗(G ) col∗(U). Let be a total order on V (G ) with C at the start, and r x 6 r s s S (G , v, ) col∗(G ) col∗(U) for every vertex v V (G ). We now define a total | r x s | 6 r x 6 r ∈ s order on V (G ) for each non-root node x V (T ). Consider each non-root node x of T x x ∈ in non-decreasing order of distance from s. Let y be the parent of x. So we may assume that is already defined. Since G is isomorphic to a subgraph of U and B B is a clique of y x x ∩ y G , there is a total order of V (G ) with B B at the start of and ordered according x x x x ∩ y x to , such that S (G , , v) col∗(G ) col∗(U) for every vertex v of G . Finally, y | r x x | 6 r x 6 r x define a total order on V (G), where v w if and only if h(v) h(w), or h(v) = h(w) ≺T and v w. Clearly is a total order on V (G). h(v) Consider w S (G, , v) for some v V (G). Thus G contains a path P = (v, w , . . . , w , w) ∈ r ∈ 1 k of length at most r with w v w for each i [k]. Let x := h(v). Since v w i ∈ i we have x h(w ), and h(w ) is not an ancestor of x. Since v, w , . . . , w is a path in T i i 1 k G, we have h(w1), . . . , h(wk) are all in the subtree of T rooted at x. By the definition of tree-decomposition, w and wk appear in a common bag. Thus h(w) is an ancestor of h(wk) or vice versa. Since w w , we have h(w) h(w ). That is, h(w) is an ancestor of h(w ), or k T k k h(w) = h(w ). Thus w B . Since w v, we have h(w) x and x is not an ancestor k ∈ h(wk) T of h(w). Hence, x is on the path from h(w) to h(w ) in T . Since w B , by the definition k ∈ h(wk) of tree-decomposition, w is in every bag in the path from h(w) to h(wk) in T . In particular, w is in Bx. This says that both endpoints of P are in Bx. If some vertex in P is not in Bx, then for some child node y of x, P contains a subpath (wi, . . . , wj), where wi, wj are distinct vertices in B B . Replace the subpath (w , . . . , w ) of P by the edge w w , which is in x ∩ y i j i j the torso Gx. Do this whenever P contains a vertex not in Bx. We obtain a path from v to w of length at most r in Gx. Since Gx is ordered by x in , we have w Sr(Gx, x, v). ∗∈ ∗ Hence S(G, , v) Sr(Gx, x, v) and S(G, , v) 6 Sr(Gx, x, v) 6 colr(Gx) 6 colr(U). ⊆ ∗ | | | | Therefore colr(G) 6 colr(U).

∗ Note that Lemma 2.14 with U = Ktw(G)+1 shows that colr(G) 6 tw(G) + 1. This implies that colr(G) 6 tw(G) + 1, as proved by Grohe et al. [87].

Lemma 2.15. For every graph G and r N, ∈ ∗ colr(G) 6 colr(G) + ω(G).

Proof. Let be a total order of V (G) such that S (G, , v) col (G) for every vertex | r | 6 r v of G. Let C be an ordered clique of G. Let 0 be the total order of V (G) obtained from by placing C at the start in the given order. For every vertex v of G, we have S (G, 0, v) S (G, , v) C, which has cardinality at most col (G) + ω(G). r ⊆ r ∪ r

We now apply König’s Lemma to show that colr is extendable.

Lemma 2.16. colr is extendable for every r N. ∈

Proof. Let G be a graph, such that for some c N for every ﬁnite subgraph H of G, we ∈ have col (H) c. Since G is countable, we may assume that V (G) = v , v ,... . Let r 6 { 1 2 }

18 Hn := G[ v1, . . . , vn ] for n N. For n N, let Vn be the set of total orders of V (Hn) { } ∈ ∈ such that Sr(Hn, , v) 6 c for every v V (Hn). By assumption, Vn = . By construction, | | ∈ S 6 ∅ V is finite. Let Q be the graph with vertex set V where in V is adjacent to − n n∈N n n n n 1 in V − if x − y if and only if x y for all vertices x, y V (H − ). So every vertex in n 1 n 1 n ∈ n 1 V has a neighbour in V − . By Lemma 2.2, there exist , ,... where is in V for n n 1 1 2 n n each n N, and n is adjacent to n−1 for each n 2. Define the relation on V (G), ∈ > where x y whenever x n y for some n N. By the definition of adjacency in Q, we have ∈ x y if and only if x n y for all n N with x, y V (Hn). Call this property (?). ∈ ∈ We now show that is a total order of V (G). Say x y and y z, where x = v and y = v i j and z = v . Let n := max i, j, k . So x, y, z V (H ). By property (?), we have x y k { } ∈ n n and y z. By the transitivity of , we have x z. By property (?), we have x z. n n n Thus is transitive. Say x y and y x, where x = v and y = v . Let n := max i, j . i j { } So x, y V (H ). By property (?), we have x y and y y. By the antisymmetry of ∈ n n n , we have x = y. Hence is antisymmetric. Consider vertices x, y V (G), where x = v n ∈ i and y = v . Let n := max i, j . So x, y V (H ). By the connexity of , we have x y j { } ∈ n n n or y x. By property (?), we have x y or y x. Hence is connex. Therefore is a n total order of V (G).

It remains to show that S (G, , v) c for each vertex v V (G). Let N be the set of | r | 6 ∈ vertices at distance at most r from v in G. So S (G, , v) N. Let n := max i : v N . r ⊆ { i ∈ } Thus S (G, , v) = S (G, , v), and S (G, , v) = S (G, , v) c, as desired. r r n | r | | r n | 6 Lemma 2.17. For every graph H and path P , and for every r N and a N0, ∈ ∈ colr((H P ) + Ka) 6 (tw(H) + 1)(2r + 1) + a.

Proof. A result of van den Heuvel and Wood [166, Lemma 30, arXiv version] implies colr(H P ) 6 (tw(H) + 1)(2r + 1) for ﬁnite H and P . By Lemma 2.16, the result holds for inﬁnite graphs. The claimed result follows by placing a copy of Ka at the start of the corresponding vertex ordering of H P .

The next lemma will be used in Section 5.5 to show that our graph that contains every X-minor-free graph has linear colouring numbers.

Lemma 2.18. Fix r N and a N0. For every graph H and path P , and for every graph ∈ ∈ G in ((H P ) + K ), D a ∗ colr(G) 6 (tw(H) + 1)(2r + 3) + 2a.

Proof. Note that ω((H P ) + Ka) = a + 2ω(H) 6 a + 2(tw(H) + 1). By Lemmas 2.15 ∗ and 2.17, colr((H P ) + Ka) 6 (tw(H) + 1)(2r + 3) + 2a. The claim then follows from Lemma 2.14.

2.7 Bounded Expansion

For r N, a graph H is an r-shallow minor of a graph G if there is a model (Xv)v∈V (H) of H ∈ in G such that each subgraph Xv has radius at most r. Nešetřil and Ossona de Mendez [131]

19 introduced the following definition (for finite graphs G). Let 2 E(H) (G) := sup | | : H is a finite r-shallow minor of G with V (H) = . ∇r V (H) 6 ∅ | | We only consider finite H in this definition, since average degree is not well-defined for infinite graphs.

A graph class has bounded expansion with bounding function f if (G) f(r) for each G ∇r 6 G and r N. We say has linear expansion if, for some constant c, for all r N, ∈ G ∈ G ∈ every graph G satisﬁes r(G) 6 cr. Similarly, has polynomial expansion if, for some ∈ G ∇ G c constant c, for all r N, every graph G satisﬁes r(G) cr . For example, when f(r) ∈ ∈ G ∇ 6 is a constant, is contained in a proper minor-closed class. As f(r) is allowed to grow with G r we obtain larger and larger graph classes.

Dvořák and Norin [68] noted that a result of Plotkin et al. [138] implies that graph classes with polynomial expansion admit strongly sublinear separators. Dvořák and Norin [68] proved the converse: A hereditary class of graphs admits strongly sublinear separators if and only if it has polynomial expansion. See [66, 69, 74] for more results on this theme.

Sergey Norin observed the following connection between colouring numbers and expansion in finite graphs; see [74]. Since every finite r-shallow minor of a graph G is a minor of a finite subgraph of G, the result for infinite graphs immediately follows.

Lemma 2.19. For every graph G and r N, ∈ (G) 2 col (G). ∇r 6 4r+1

2.8 Planar Triangulations

This subsection introduces some elementary notions regarding planar graphs that form a foundation for the results in Section3.A plane graph is a graph embedded in the plane, i.e. drawn without crossings. A near-triangulation is a 2-connected ﬁnite plane graph, in which each face, except possibly one, is bounded by a 3-cycle. The exceptional face is bounded by the outer cycle.

A graph G is locally Hamiltonian if for each vertex v V (G), the subgraph G[N (v)] has ∈ G a Hamiltonian cycle. By deﬁnition, every locally Hamiltonian graph is locally ﬁnite. It is easily seen that every connected, locally Hamiltonian graph is 3-connected.

A plane triangulation is a connected locally Hamiltonian plane graph G such that for each vertex v V (G), there is a Hamiltonian cycle C of G[N (v)] such that v is the only vertex ∈ v G of G inside Cv or v is the only vertex of G outside Cv. The subgraph of G consisting of v, Cv, and the edges from v to Cv form a wheel centred at v.A planar triangulation is a planar graph that can be embedded in the plane as a plane triangulation. A planar triangulation G is a triangulation of a planar graph H if H is a spanning subgraph of G. Lemma 2.20. Every connected locally Hamiltonian plane graph G is a plane triangulation.

Proof. For each vertex v V (G), let C be a Hamiltonian cycle through N (v). Let ∈ v G v V (G) and let pq E(C ). So vpq is a triangle in G. Since C p q is connected, ∈ ∈ v v − −

20 every vertex in C p q is inside vpq or every vertex in C p q is outside vpq. In v − − v − − both cases, vp and vq are consecutive in the cyclic ordering of edges incident to v defined by the embedding. Thus the cyclic ordering of NG(v) defined by Cv coincides with the cyclic ordering of NG(v) defined by the embedding. Without loss of generality, v is inside Cv. Suppose for a contradiction that some other vertex x is inside Cv. So x is inside the triangle vpq for some pq E(C ). Choose such a vertex x to minimise the distance in G from x to ∈ v v, p, q . Since vp and vq are consecutive in the cyclic ordering of edges incident to v, v has { } no neighbour inside vpq. In particular, vx E(G). Since G is connected and by the choice 6∈ of x, without loss of generality, px E(G). Let y be the neighbour of p inside vpq, such that ∈ pv and py are consecutive in the cyclic ordering of edges incident to p (which is well-defined since x is a candidate). Since Cp coincides with the cyclic ordering of neighbours of p, we have vy E(G), which contradicts the fact that v has no neighbour in vpq. Hence, v is the ∈ only vertex inside Cv. And G is a planar triangulation.

By deﬁnition, a planar triangulation is locally Hamiltonian and is thus 3-connected (as observed above). This can be strengthened as follows.

Lemma 2.21. Every ﬁnite subgraph of a planar triangulation G can be extended to a ﬁnite 3-connected subgraph of G.

Proof. Let H0 be a finite subgraph of G. Add shortest paths between the components of H0 to obtain a finite connected subgraph H1 of G. Let H be obtained from H1 by adding the wheel in G centred at each vertex in H1. Then H is a finite 3-connected subgraph of G.

This lemma is in sharp contrast to the fact that there exist planar graphs of arbitrarily large (ﬁnite) connectivity such that no ﬁnite subgraph is 3-connected4.

A locally finite graph G is k-ended if k is the maximum integer such that G S has k infinite − components for some finite set S V (G). ⊆

3 Forcing a Minor or Subdivision

This section proves out ﬁrst main result, Theorem 1.1, which says that if a graph U contains

every planar graph, then (a) the complete graph Kℵ0 is a minor of U, and (b) U contains a subdivision of Kt for every t N. The proof is split across three subsections. In Section 3.1 ∈ we introduce the notion of a ‘limit’, which may be of independent interest, and we prove the ‘Limit Lemma’ (Lemma 3.1), which shows that any graph that contains uncountably many planar graphs of a certain type contains a planar triangulation along with an inﬁnite number of ‘jump’ paths. Then Section 3.2 presents a number of lemmas about routing paths in graphs obtained from planar triangulations by adding jumps. All of these results are then combined in Section 3.3, where the proof of Theorem 1.1 is completed. In fact, we prove

4 For example, let Gk be a planar graph containing cycles C0,C1,... drawn as concentric circles in the plane, such that for i ∈ N0 each vertex in Ci is adjacent to at least k vertices in Ci+1, and for i ∈ N each vertex in Ci is adjacent to at most one vertex in Ci−1 and some vertex in Ci is adjacent to no vertex in Ci−1.

Then Gk is (k + 2)-connected but contains no 3-connected ﬁnite subgraph.

21 signiﬁcant strengthenings of both parts of Theorem 1.1. Finally, in Section 3.4 we show that numerous graph classes do not have a universal element, including k-apex graphs, chordal graphs containing no Kℵ0 , graphs with no Kℵ0 minor (which was proved in [56]), and graphs containing no Kℵ0 subdivision (which was left open in [56]).

3.1 Limit Lemma

Let G be a countably infinite graph. Let be an uncountable set of infinite, locally finite, G connected subgraphs of G.A -limit in G is any subgraph of G that can be obtained as G follows. Let v0 be any vertex in G that is contained in uncountably many distinct subgraphs in (which exists since is uncountable and G is countable). Denote this subset of by G G G . Define an equivalence relation on , where two graphs in are equivalent if they G0 G0 G0 contain the same set of edges incident to v . By the choice of v and since every graph in 0 0 G is connected, this set of edges is nonempty. The number of equivalence classes is countable since every graph in is locally finite. Since is uncountable, some equivalence class is G G0 uncountable. Call this equivalence class . Let v , v ,... be the remaining vertices of G. G1 1 2 We now define a sequence of uncountable families ... . Suppose that G0 ⊇ G1 ⊇ G2 ⊇ Gn is defined for some n N. Define an equivalence relation on n, where two graphs in n ∈ G G are equivalent if they contain the same set of edges incident to vn (which may be empty). Again, the number of equivalence classes is countable since every graph in is locally finite. G Let be an uncountable equivalence class. Since every graph in is connected, v is Gn+1 G n contained in all or none of the graphs in . Let H be the graph with vertex set V (G) Gn+1 such that for every n > 0, the set of edges incident with vn in H is the set of edges incident with v in each graph in . We call H a -limit in G. n Gn+1 G By construction of -limits, each finite induced subgraph of H is an induced subgraph of G every graph in for some sufficiently large n. This implies that H inherits many properties Gn of (in the spirit of extendable properties introduced in Section 2.1). For example, by G Lemma 2.1, if every graph in is planar, then H is planar. Every degree of a vertex in H is G also the degree of some vertex of a graph in . However, there is one important property G that need not be preserved, namely connectedness. To see this, let G be obtained from the infinite ladder K2 C∞ by replacing each edge by two internally disjoint paths of length 2, and let be the set of 2-way infinite paths of G. Then the -limits in G are precisely G G those spanning subgraphs that are either a 2-way infinite path plus infinitely many isolated vertices, or the union of two disjoint 2-way infinite paths. Therefore we shall focus on a connected component of a limit. Some components may consist of isolated vertices. It is easy to see that every finite component of a limit graph (if any) is trivial (that is, an isolated vertex). But, the component containing v0 does contain edges and is therefore infinite. The property of being 1-ended need not be preserved under taking limits. For example, if G is the 2-way infinite path with edges doubled and is the set of all 1-way infinite paths of G G, then the -limits in G are all 1-way paths (together with isolated vertices) and all 2-way G infinite paths, and the latter are 2-ended.

Therefore we introduce the following stronger property: for a function f : N N, a plane → graph G is f-isoperimetric if G has at most f(n) vertices inside every cycle of length n.A

22 class of plane graphs is f-isoperimetric if every graph in is f-isoperimetric. A planar G G triangulation G is f-isoperimetric if G has an f-isoperimetric embedding in the plane. For example, every triangulation of the infinite grid C∞ C∞ is f-isoperimetric for some f with f(n) O(n2); see [163]. As another example, it follows from Euler’s formula that ∈ every plane triangulation with minimum degree at least 7 is f-isoperimetric for some f with f(n) O(n). ∈ Note that every f-isoperimetric infinite planar triangulation G is 1-ended. To see this, suppose for the sake of contradiction that G S has at least two infinite components for some − finite S V (G). As noted earlier, G contains a finite 3-connected subgraph H containing ⊆ S. Since G V (H) has at least two infinite components, at least two faces of H contain − infinitely many vertices of G. One of these faces is the interior of a cycle C of length, say n, in H. But this contradicts that G has at most f(n) vertices inside C.

Let H be a subgraph of a graph G.A jump of H in G is a path P in G such that the endpoints of P are non-adjacent distinct vertices in H and no internal vertex of P is in H. (A jump can be a single edge.)

Lemma 3.1 (Limit Lemma). Fix an arbitrary function f : N N. Let G be a countably → inﬁnite graph. Let be an uncountable set of subgraphs of G, each of which is a 4-connected G f-isoperimetric planar triangulation. Let L be any non-trivial connected component of some -limit in G. Then L is a 4-connected f-isoperimetric planar triangulation, and there are G inﬁnitely many pairwise disjoint jumps of L in G.

Proof. Let L be a -limit in G defined with respect to a sequence of vertices v , v ,... and 0 G 0 1 a sequence of families ... such that L is a non-trivial connected component of L . G0 ⊇ G1 ⊇ 0 By construction, every finite induced subgraph of L is an induced subgraph of uncountably many graphs in . Since every graph in is planar, every subgraph of L is planar, and G G L is planar by Lemma 2.1. By the construction of a -limit, for every vertex v of L, for G some G0 , the subgraph of L induced by N [v] is equal to the subgraph of G0 induced by ∈ G L NG0 [v]. So L is locally Hamiltonian. By Lemma 2.20, L is a planar triangulation. Assume that L is drawn in the plane so that the vertex set is unbounded5. We now show this embedding of L is f-isoperimetric. Suppose that C is a cycle of length k in L, and L has more than f(k) vertices inside C. Let C0 consist of C along with f(k) + 1 vertices inside C and also f(k) + 1 vertices outside C (which exist since L is drawn so that the vertex set is unbounded). By Lemma 2.21, C0 is contained in a finite 3-connected subgraph C00 of L, which is a subgraph of some triangulation T in . Since C00 is 3-connected, it has a unique G embedding in the plane up to the choice of the outer face. Thus T is a plane triangulation in with more than f(k) vertices inside C (and outside C), which contradicts that is G G 5We now prove that every infinite planar graph can be drawn in the plane so that the vertex set is unbounded. If the vertex set is bounded, then the graph has an accumulation point p. We may think of the graph being drawn on the sphere with p as the north pole. Deleting p from the sphere results in the graph drawn in the plane. A sequence of vertices converging to p on the sphere now becomes a sequence of vertices tending to infinity in the plane. Note that, if p is a vertex or is a point on an edge of the graph, then the graph can be redrawn, vertex by vertex, such that p is never used and such that each new vertex in the new embedding is close to the vertex in the old embedding so that the vertex-accumulation points are the same. We refer to [127] for further results about infinite graphs on surfaces.

23 f-isoperimetric. Hence L is f-isoperimetric. Since every graph in is 4-connected, we may G assume that f(3) = 0. Since L is locally Hamiltonian, every separating set with three vertices in L induces a triangle. Since L is f-isoperimetric and f(3) = 0, every triangle in L is a face, implying that L is 4-connected.

It remains to prove that L has infinitely many pairwise disjoint jumps in G. To prove this, it suffices to show that for all pairwise disjoint jumps J1,...,Jp of L in G, there exists Jp+1 such that J1,...,Jp+1 are pairwise disjoint jumps of L in G. Let H be the subgraph of L induced by the end-vertices of J1,...,Jp. Let W be the set of internal vertices in J1,...,Jp. So W V (G) V (L). ⊆ \ Define the following finite subgraph H0 of G disjoint from L. Consider each component 0 0 T of L0 that intersects W . If V (T ) = 1 then add this vertex as an isolated vertex to 0 0 | | 0 0 H . Otherwise, T is a planar triangulation. Let C1(T ) be a cycle in T whose interior 0 0 0 0 0 contains W V (T ). Let C2(T ) be a cycle in T such that C1(T ) is inside C2(T ), and ∩ 0 0 each vertex of C2(T ) is adjacent to a vertex of C1(T ). (Note that “inside" is well-defined 0 0 since T is 1-ended.) The near-triangulation consisting of C2(T ) and all the vertices and 0 0 0 edges of T inside C2(T ) is called the protecting near-triangulation of T (as it, intuitively, “protects" those vertices of W that are in T 0). Add each such protecting near-triangulation as a component of H0.

0 0 Since H H is finite, V (H H ) v1, v2, . . . , vn for some sufficiently large n. Since ∪ ∪ ⊆ { } 00 n+1 has uncountably many triangulations, n+1 contains at least one triangulation, say T , G 00 G distinct from L. We claim that T contains the required additional jump Jp+1. To prove the claim, first observe that T 00 contains an edge not in L (since it is easy to prove that a 4-connected triangulation cannot contain a triangulation as a proper subgraph). Since T 00 is connected and intersects L, T 00 has an edge uw not in L such that u is in L. Since T 00 and L agree on the set of edges in G incident with each vertex in H, we have that u is not in H. Also, w is not in H because L and T 00 agree on the edges incident to vertices in H.

If w is in L, then we may take Jp+1 = uw (since neither u nor w is in H), as claimed. Now assume that w is not in L.

Since T 00 is 4-connected, by Menger’s theorem, T 00 u has three paths P ,P ,P from w − 1 2 3 to L which are pairwise disjoint except that they all contain w. Let P4 be the edge wu. Each of P ....,P have exactly one end-vertex in L. Observe that P P P P does 1 4 1 ∪ 2 ∪ 3 ∪ 4 not intersect H by the same argument that u is not in H. In particular, no end-vertex of J ,...,J is in P P P P . 1 p 1 ∪ 2 ∪ 3 ∪ 4 Claim 3.1.1. If xy is an edge in P P P P and x is in some protecting near- 1 ∪ 2 ∪ 3 ∪ 4 triangulation N and y is not in N, then x is on the outer-cycle of N.

Proof. Towards a contradiction suppose x is not on the outer-cycle of N. Then the wheel centred at x in N is also the wheel in T 00 centred at x, by the choice of n. In particular, y is also a neighbour of x in N, which is a contradiction.

Note that if x is an isolated vertex in H0, then x is isolated in L and in T 00 (since L and 00 0 00 T agree on edges incident to vertices in H ). Since P1,P2,P3,P4 are in T , no vertex in

24 P P P P is isolated in T 00. Thus no vertex in P P P P is isolated in H0. In 1 ∪ 2 ∪ 3 ∪ 4 1 ∪ 2 ∪ 3 ∪ 4 particular, w is not isolated in H0. It is possible that w is in H0, in which case w is on the outer cycle of one of the protecting near-triangulations by Claim 3.1.1.

Let N be a protecting near-triangulation such that V (N) W V (P P P ) = . For each ∩ ∩ 1 ∪ 2 ∪ 3 6 ∅ i [3], let x be the ﬁrst vertex of P (beginning from w) in N, and let y be the last vertex ∈ i i i of P that is in N. Note that if V (P ) V (N) = 1, then x = y , and if V (P ) V (N) = , i | i ∩ | i i i ∩ ∅ then both x and y are undeﬁned. Let X := x , x , x and Y := y , y , y . Let C be i i { 1 2 3} { 1 2 3} the outer cycle of N, D be the outer cycle of N C, and N 0 := N[V (C) V (D)]. By − ∪ Claim 3.1.1, X Y V (C). Moreover, by construction, W V (N) is disjoint from V (N 0). ∪ ⊆ ∩ Since N 0 is 3-connected, there are three vertex-disjoint paths from X to Y in N 0. Therefore, by rerouting, we may assume that V (N) W V (P P P ) = . ∩ ∩ 1 ∪ 2 ∪ 3 ∅ Repeating the above argument for each protecting near-triangulation N such that V (N) ∩ W V (P P P ) = , we may assume that P P P P is disjoint from W . Since L ∩ 1 ∪ 2 ∪ 3 6 ∅ 1 ∪ 2 ∪ 3 ∪ 4 is 4-connected, it contains no K . So there are distinct i, j [4] such that the end-vertices of 4 ∈ P and P in L are non-adjacent in L. Thus, we may take J = P P , which completes i j p+1 i ∪ j the proof.

3.2 Routing with Jumps

This subsection proves a series of lemmas about routing paths in graphs obtained by adding jumps to planar triangulations. These results are used in the proofs of Theorems 3.9 and 3.11 below.

Consider the grid graph G = P` Pm, where P` is the path (x1, . . . , x`) and Pm is the path (y , . . . , y ). For i [`], the subpath Ci := ((x , y ),..., (x , y )) is called the i-th column of 1 ` ∈ i 1 i m G.

Lemma 3.2. Let G = P` Pm for some `, m N. Let a1, . . . , ak be vertices (ordered from 1 ∈ top to bottom) in the ﬁrst column C of G, and let b1, . . . , bk be vertices (ordered from top to m 1 m bottom) in the last column C of G. If ` > 2k + 1, then there exist k vertex-disjoint C –C paths P 1,...,P k in G such that the end-vertices of P i are a and b for each i [k]. i i ∈

Proof. We proceed by induction on k > 0. The case k = 0 is vacuous. By symmetry, we may 1 assume that b1 is above a1. Let P be the path starting at a1 which proceeds one unit to i the right, up to the row containing b1, and then right until it reaches b1. For i [2, k], let Q ∈ 3 be the path starting at ai and proceeding two units to the right to the third column C of G, and let a0 be the end-vertex of Qi on C3. By induction, there exist k 1 vertex-disjoint i − C3–Cm paths S2,...,Sk in G V (P 1) such that the end-vertices of Si are a0 and b for − i i each i [2, k]. Deﬁning P i := Qi Si for each i [2, k] completes the proof. ∈ ∪ ∈

The next lemma is a special case of the 2-linkage theorem, independently due to Thomassen [158] and Seymour [151]. For a graph G and u, v V (G), let G + uv be ∈ the graph obtained from G by adding the edge uv if it does not already exist. Lemma 3.3. Let G be a plane near-triangulation with outer-face C, p , q , p , q V (C), { 1 1 2 2} ⊆ and u and v be non-adjacent vertices of G such that neither u nor v is on C. If G does

25 not contain a ( 3)-separation (G ,G ) such that p , q , p , q V (G ) and G is planar, 6 1 2 { 1 1 2 2} ⊆ 1 1 then there are vertex-disjoint paths P and Q in G + uv such that the end-vertices of P1 are p1 and p2 and the end-vertices of Q are q1 and q2.

For a cycle C in a plane graph G, let ∆(C) R2 be the closed disk bounded by C. We say ⊆ that C is tight if there is no cycle D in G such that ∆(C) ∆(D) and V (D) < V (C) .A ⊆ | | | | cycle D in G surrounds a cycle C in G if V (C) V (D) = and ∆(C) ∆(D). ∩ ∅ ⊆ Lemma 3.4. For every cycle C of an inﬁnite f-isoperimetric plane triangulation G, there is a tight cycle that surrounds C.

Proof. Let be the set of cycles in G with vertex set V (C). (Chords of C can create cycles C in distinct from C.) Since G is locally finite, is finite. Let V be the set of vertices of G C C in S ∆(D). Since G is f-isoperimetric, V is finite. Let C1 be the facial cycle of G V D∈C − bounding V . So C1 surrounds C. If C1 is tight, then C1 satisfies the claim. Otherwise, there is a cycle C2 with ∆(C1) ∆(C2) and V (C2) < V (C1) . So C2 surrounds C. If C2 is tight, ⊆ | | | | then C2 satisfies the claim. Repeating this argument we obtain a sequence C1,C2,... of non-tight cycles with ∆(Ci) ∆(Ci+1) and V (Ci+1) < V (Ci) , where each Ci surrounds ⊆ | | | | C. Say Ci has length ` and n vertices of G are in ∆(Ci). Since V (Ci+1) < V (Ci) , we i i | | | | have Ci = Ci+1 and n > n . So ` ` (in fact, ` ` i + 1, but we will not use that). 6 i+1 i i 6 1 i 6 1 − Moreover, n + i 1 n f(` ) f(` ) (since we may assume that f is a non-decreasing 1 − 6 i 6 i 6 1 function). With i = f(`1) + 1, we have n1 6 0, which is a contradiction. Lemma 3.5. Let G be a plane graph, let C be a tight cycle in G, and let D be a cycle in G such that ∆(C) ∆(D). Then there are V (C) vertex disjoint paths in G from V (C) to ⊆ | | V (D).

Proof. Suppose that the desired paths do not exist. By Menger’s Theorem, there exists X V (G) with X < V (C) such that there is no path in G X between V (C) and V (D). ⊆ | | | | − Since G is a triangulation, there is a cycle C0 between C and D with V (C0) = X. But now, C0 contradicts that C is tight.

Repeatedly applying Lemma 3.5 we obtain the following. Lemma 3.6. Let C1,...,Cm be tight cycles in a plane graph G such that V (C1) = `, and j i | | C surrounds C for all 1 6 i < j 6 m. Then G contains a subdivision of C` Pm, where i the branch vertices of the i-th cycle of C` Pm are in C . Lemma 3.7. Let G be a 4-connected f-isoperimetric plane triangulation, and let C be a tight cycle in G of length ` > 2k + 4. Let x1, . . . , xk be vertices in clockwise order around C. Then there is a tight cycle D surrounding C, and there are vertex-disjoint paths Q1,...,Qk from C to D such that:

• x is the unique vertex of Qi on C for each i [k], i ∈ • if y is the other end-vertex of Qi for each i [k], then y , y , y , . . . , y appear in this i ∈ 2 1 3 k clockwise order on D, • Q1 Qk is contained in the subgraph of G between C and D together with one ∪ · · · ∪ edge e M. ∈

26 Proof. By Lemma 3.4, there are tight cycles C = C1,C2,...,C2k+1 such that Cj surrounds i C for all 1 6 i < j 6 2k + 1; see Figure3. Since G is f-isoperimetric, there are only ﬁnitely many vertices of G inside C2k+1. By Lemma 3.4 and since M is inﬁnite, there is a tight cycle C2k+2 surrounding C2k+1 and there is an edge vw M such that v ∆(C2k+2), and ∈ ∈ v, w ∆(C2k+1) = . By Lemma 3.4, there exists a tight cycle C2k+3 surrounding C2k+2 { } ∩ ∅ such that w ∆(C2k+3) and w / V (C2k+3). Let C2k+4 be a tight cycle surrounding C2k+3 ∈ ∈ with ∆(C2k+4) minimal (under inclusion).

y1 y2

Pb Pa z1 Pb+1 z2 Pa 1 − x1 x2

C2k+4

C2k+3 x3 b b b C3 C2 C1 C2k+2 xk C2k+1

x6 x4 x5

z6 yk y3 z4 z5

Figure 3: Illustration for the proof of Lemma 3.7.

By Lemma 3.6, G contains a subgraph H that is a subdivision of C` P2k+4, where the i branch vertices of the i-th cycle of C` P2k+4 are contained in C . Choose such an H with V (H) minimum. Let P 1,...,P ` be the C1–C2k+4 paths of the subdivision (ordered | | clockwise). For i, j [`], let G(i, j) be the subgraph of G induced by the vertices between ∈ C2k+1 and C2k+4 (inclusive) and between P i and P j (inclusive and clockwise from P i to P j). Since ` 2k + 4, there exist a, b [`] such that v and w are both in G(a, b), and there > ∈ are at least k branch vertices of H in V (C2k+1) V (G(a, b)). By symmetry, we may assume \ 1 < a < b < `. For each i [`], let s and t be the unique vertices of P i on C2k+1 and C2k+4, ∈ i i

27 2k+1 respectively. Let z1, z2, . . . , zk be ordered clockwise on C where z1 := sa−1, z2 := sb+1, and z , . . . , z are branch vertices of H contained in V (C2k+1) V (G(a 1, b + 1)). Let 3 k \ − H0 := H (V (C2k+2) V (C2k+3) V (C2k+4)). Note that H0 is a subdivision of C P , − ∪ ∪ ` 2k+1 which is a supergraph of P` P2k+1. Therefore, by Lemma 3.2 there exist vertex-disjoint C1–C2k+1 paths R1,...,Rk in H0 such that the end-vertices of Ri are x and z for all i [k]. i i ∈ For i [3, k] we use H to extend Ri to a path Qi with an end-vertex y in C2k+4. Let ∈ i y1 := ta−1 and y2 := tb+1. Below we prove there exists vertex-disjoint paths Q and Q in G(a 1, b + 1) + vw such 1 2 − that the end-vertices of Q1 are z1 and y2 and the ends of Q2 are z2 and y1. Use Q1 to extend 1 1 2 2 R to a path Q with y2 as an end-vertex, and use Q2 to extend R to a path Q with y1 as an end-vertex, as desired. It remains to prove the existence of the paths Q1 and Q2 in G(a 1, b + 1) + vw. − By construction, neither v nor w is on the outer-cycle O of G(a 1, b+1). We claim that G(a − − 1, b+1)+vw does not contain a ( 3)-separation (G ,G ), such that s − , t − , s , t 6 1 2 { a 1 a 1 b+1 b+1} ⊆ V (G1) and G1 is planar. Suppose for the sake of contradiction that (G1,G2) is such a separation. Since G is 4-connected, (G1,G2) is a 2- or 3-separation. First suppose it is a 2-separation, say V (G1) V (G2) =: x, y . Since G is a plane triangulation, xy is a ∩ { } 2k+4 chord of O. Since s − , t − , s , t V (G ), we have x, y V (C ), x, y { a 1 a 1 b+1 b+1} ⊆ 1 { } ⊆ { } ⊆ V (P a−1), x, y V (P b+1), or x, y V (C2k+1). Since C2k+1 is tight, x, y V (C2k+1). { } ⊆ { } ⊆ { } 6⊆ By the minimality of V (H) , x, y V (P a−1) and x, y V (P b+1). By the minimality | | { } 6⊆ { } 6⊆ of ∆(C2k+4), x, y V (C2k+4). Thus, we may assume that (G ,G ) is a 3-separation, { } 6⊆ 1 2 say V (G ) V (G ) =: x, y, z . Since G is 4-connected, every component of (G(a 1, b + 1 ∩ 2 { } − 1) + vw) x, y, z must contain a vertex of O. It follows that x, y, z V (O) 2. By − { } |{ } ∩ | > symmetry, we may assume that x, z V (O). If xz is a chord of O, then G cannot contain { } ⊆ 1 all of sa−1, ta−1, sb+1, tb+1 by the previous case. Thus, xz is not a chord of O. If y is not adjacent to both x and z, then G x, z is a near-triangulation, and hence G x, y, z is − { } − { } connected. Thus, xyz is a path of G.

2k+4 Since s − , t − , s , t V (G ), me must have x, z V (C ), x, z { a 1 a 1 b+1 b+1} ⊆ 1 { } ⊆ { } ⊆ V (P a−1), x, z V (P b), or x, z V (C2k+1). If x, z V (P a−1), x, z V (P b+1), or { 2k}+4 ⊆ { } ⊆ { } ⊆ 0 { } ⊆ 0 x, z V (C ), then G1 is non-planar since it is of the form G1 + vw, where G1 is a near { } ⊆ 0 triangulation and at least one of v or w is not on the outer-face of G1. The remaining case is x, z V (C2k+1). Since C2k+1 is tight, there must be a vertex p such that zp E(G), { } ⊆ ∈ and x, p, z are consecutive vertices of O. However, since v, w x, p, z = , G is again { } ∩ { } ∅ 1 non-planar.

Therefore, by Lemma 3.3, there are vertex-disjoint paths Q and Q in G(a 1, b + 1) + vw 1 2 − such that the end-vertices of Q1 are sa−1 and tb+1 and the end-vertices of Q2 are sb+1 and ta−1, as required.

The next lemma is a key ingredient in the proof of Theorem 1.1(a).

Lemma 3.8. Let G be an f-isoperimetric 4-connected plane triangulation, for any function f : N N. Let Q be a graph obtained from G by adding an inﬁnite matching M of edges → not in G. Then Kℵ0 is a minor of Q.

28 Proof. Say (a, b) (c, d) if a < c, or a = c and b d. We claim that for every i N there 6 6 ∈ exists a 1-way infinite path Pi in G, and for every j N with i 6 j, there exists a tight cycle i ∈ Dj in G satisfying the following: k i • D` surrounds Dj if (j, i) 6 (`, k), 1 • The first vertex of Pi is on D for each i N, i ∈ • V (P ) V (Dj ) = 1 for all i, j k, | i ∩ k | 6 • For all i < j, there exists ` max j, k such that the two vertices of P P on Dk > { } i ∪ j ` are consecutive in the cyclic ordering of the i vertices of P P on Dk. 1 ∪ · · · ∪ i ` The last condition implies that for all i < j, there is Pi–Pj path Pi,j in G such that all these paths are pairwise vertex-disjoint. Therefore, we obtain a Kℵ0 minor by contracting each Pi to a vertex and contracting all but one edge from each Pi,j. So, it suffices to prove the above claim.

1 1 Let D1 by any tight cycle in G. Let the first vertex of P1 be any vertex of D1. Fix k > 2, i and suppose that Dj has been defined for all j < k and i 6 j, and that the vertices of Pi in k−1 1 1 ∆(Dk−1) have been defined for all i < k. Let Dk be any tight cycle such that Dk surrounds Dk−1 and V (D1) 2k + 4. Note that D1 exists by repeatedly applying Lemma 3.4 and k−1 | k | > k using the fact that G is f-isoperimetric. By induction, for all i < j k 1, there exist j ` 6 − 6 and k ` such that the two vertices of P P on Dk are consecutive in the cyclic ordering 6 i ∪ j ` of the i vertices of P P on Dk. 1 ∪ · · · ∪ i ` 1 By Lemma 3.5, we can extend P1,...,Pk−1 so that each path has one end-vertex in Dk. 1 Let X be the set of end-vertices of P1,...,Pk−1 on Dk. Choose an arbitrary vertex a of V (D1) X to be the first vertex of P . Suppose the cyclic ordering of X a on D1 is k \ k ∪ { } k x1, . . . , x`, a, x`+1, . . . , xk. By Lemma 3.7, there is a tight cycle D such that D surrounds 1 Dk. Moreover, we can extend P1,...,Pk so that each path has one end-vertex in D, and the 2 cyclic ordering of X a on D is x , . . . , x − , a, x , . . . , x . Define D := D. Repeat this ∪ { } 1 ` 1 ` k k argument to define D3,...,Dk. By construction, for all i [k 1], x and a are consecutive k k ∈ − i in the cyclic ordering of X a on Dj for some j k, as required. ∪ { } k 6

3.3 Proof of Theorem 1.1

We now combine the lemmas from the previous subsections to prove our ﬁrst main theorem.

Theorem 3.9. Let U be a countable graph containing an uncountable family of subgraphs, G each of which is a 4-connected f-isoperimetric planar triangulation, for an arbitrary function f : N N. Then the complete graph Kℵ0 is a minor of U. →

Proof. Let L be any non-trivial connected component of any -limit in U. By Lemma 3.1, G L is a 4-connected f-isoperimetric planar triangulation, and there exists an inﬁnite set of P pairwise-disjoint jumps of L in U. Let Q be the graph obtained from U by contracting each path in to an edge, and deleting any remaining vertices not in L and not in a jump in . P P Observe that Q is isomorphic to a graph obtained from L by adding an inﬁnite matching of edges not in L. By Lemma 3.8, Kℵ0 is a minor of Q. The result follows since Q is a minor of U.

29 Let G := C∞ C∞ be the inﬁnite grid. Every triangulation of G is f-isoperimetric where f(n) O(n2). There are uncountably many triangulations of G, since in each face there ∈ are two possible edges to add. Since G has only countably many automorphisms, any uncountable set of triangulations of G contains an uncountable subset of pairwise non- isomorphic triangulations of G. Thus Theorem 3.9 implies and strengthens Theorem 1.1(a). Indeed, since there are uncountably many 6-regular triangulations of G, we obtain the following stronger result6.

Theorem 3.10. If a graph U contains every 6-regular triangulation of the inﬁnite grid, then

Kℵ0 is a minor of U.

For each t N, let Kt be the multigraph on t vertices with t edges between every pair of ∈ t vertices. The next theorem implies and strengthens Theorem 1.1(b).

Theorem 3.11. If a graph U contains every planar graph, then U contains a subdivision of Kt for every t N. t ∈

2 Proof. Let m 2(t 1)t + 4. Let H be a planar graph obtained from C P∞ by > − m adding t new vertices w , . . . , w , such that w is adjacent to t(t 1) vertices of the ﬁrst 1 t i − cycle of Cm P∞, and NH (wi) NH (wj) = for all distinct i, j [t]. Let f : N N be ∩ ∅ ∈ → such that every triangulation of H is f-isoperimetric. Choose ` N so that ` 2m and ∈ > `m 2f(m 1). > − Let Y be the set of vertices in H contained in the ﬁrst ` cycles of Cm P∞ together with + w1, . . . , wt. Let H be obtained from H by adding some edges with both end-vertices in Y so that H+[Y ] is a 4-connected planar triangulation. Finally, let be the family of all F 4-connected triangulations of H+.

Since U contains all planar graphs, for each F , there is an injective homomorphism ∈ F φF : V (F ) V (U). Since is uncountable, Y is ﬁnite, and V (U) is countable, there exists → F 0 an uncountable subset of such that φ (y) = φ 0 (y) for all F,F and all y Y . F0 F F F ∈ F0 ∈ Let be the collection of subgraphs G of U such that there exists an F such that φ−1 G ∈ F0 F is an isomorphism from G to F .

Choose F and let X := φ (Y ) (note that X is independent of the choice of F ). Let L0 be ∈ F0 F a -limit in U, with respect to an enumeration x , x ,... of V (U), where X = x , . . . , x . G 1 2 { 1 |X|} Let L be the component of L0 which contains X. Thus L[X] is isomorphic to H+[Y ]. Moreover, by Lemma 3.1, L is a 4-connected f-isoperimetric triangulation, and there is an inﬁnite set of pairwise disjoint jumps of L in U. For i [`], let C` be the cycle in L J + ∈ corresponding to the `-th cycle of Cm P∞ in H . Claim 3.11.1. C1 is a tight cycle of L.

Proof. Let D be a cycle in L such that ∆(C) ∆(D). If V (D) X, then V (D) V (C) ⊆ ⊆ | | > | | since L[X] is isomorphic to H+[Y ]. Thus, we may assume that D contains a vertex x / X. ∈ 6 If cx ∈ {+, −} for each x ∈ Z, then adding the edge with slope cx across every face of the infinite grid with left x-coordinate x, gives a 6-regular triangulation of the infinite grid. Since there are uncountably many choices for (cx)x∈Z there are uncountably many 6-regular triangulations of the infinite grid.

30 d ` e If D contains a vertex of C 2 , then V (D) > m = V (C) , since ` > 2m. Thus, we may | |d ` e | | ` assume that D does not contain a vertex of C 2 . Therefore, there are at least m vertices d 2 e of L inside D. Since `m 2f(m 1) and L is f-isoperimetric, we have that V (D) m. > − | | >

t t t Let D(Kt ) be a drawing of Kt such that exactly two edges of Kt intersect at each crossing, 0 0 and the number of crossings is ﬁnite, say c. Let w1, . . . , wt be the vertices of L corresponding 0 0 to the vertices w , . . . , w of H. Let u , . . . , u 2 be the set of neighbours of w , . . . , w 1 t 1 (t−1)t { 1 t} on C1. Recall that L is a 4-connected f-isoperimetric plane triangulation, C1 is a tight cycle in L, and V (C1) 2(t 1)t2 + 4. Applying Lemma 3.7 c times, there exist tight | | > − cycles D1 := C1,D2,...,Dc+1 of L, and jumps J ,...,J in , and a subdivision K of Kt 1 c J t in L J J such that ∪ 1 ∪ · · · ∪ c 0 0 • w1, . . . , wt are the branch vertices of K, • J1 Jc K, and ∪ · · · ∪ ⊆ 2 i i+1 • each jump Ji can be drawn in the region of R between D and D to obtain a t t drawing of Kt equivalent to D(Kt ).

3.4 Excluding Complete Graph Subdivisions

As mentioned earlier, Diestel et al. [56] proved that, for each integer t > 5, there is no universal graph for the class of graphs having no Kt-minor, which is an immediate consequence of our Theorem 1.1(a). They also proved that for each integer t > 5, there is no universal graph for the class of graphs containing no Kt-subdivision. We now show that this result can be proved using the Limit Lemma (Lemma 3.1). A graph is k-apex if it contains a set (called the apex set) of at most k vertices whose deletion results in a planar graph.

Lemma 3.12. If a graph G contains all k-apex graphs, then G contains a subdivision of

Kk+5.

Proof. Let be an uncountable family of 4-connected f-isoperimetric plane triangulations, F for any function f : N N. For each set A of k vertices in G, let A be the set of subgraphs → G H of G such that A is a clique in H, A is complete to H A, and H A is isomorphic to a − − graph in . Since G contains all k-apex graphs and there are only countably many such sets F A, is uncountable for some set A. Let 0 := H A : H . Let L be any non-trivial GA GA { − ∈ GA} component of any 0 -limit in G. By Lemma 3.1, L is a 4-connected f-isoperimetric planar GA triangulation, and there are inﬁnitely many pairwise disjoint jumps of L in G. Since A is ﬁnite, one of these jumps, say J, is disjoint from A. Let x and y be the end-vertices of J. Since L is 4-connected, there is a cycle C in L such that V (C) 4 and x, y V (C). | | > { } ⊆ Let L be the subgraph of L contained in the closed disk bounded by C. Note that L J C C ∪ contains a subgraph K which is a subdivision of K4. Since L is 4-connected, K can be extended to a subdivision of K in L J. Together with A, we obtain a subdivision of K 5 ∪ k+5 in G.

Since no k-apex graph contains a subdivision of Kk+5, Lemma 3.12 implies:

31 Corollary 3.13. For each k N, there is no universal graph for the class of k-apex graphs, ∈ and, for each integer k > 5, there is no universal graph for the class of graphs containing no subdivision of Kk.

The class of graphs containing no subdivision of Kℵ0 is not covered by Corollary 3.13. We now show there is no universal graph for this class, amongst many other examples.

Lemma 3.14. Let be a non-empty class of graphs such that no graph in contains Kℵ , G G 0 and for every graph G and for every vertex v V (G), the graph G v contains a ∈ G ∈ − subgraph in . Let be the family of graphs that contain no subgraph in . Then there is G F G no universal graph for . F

0 Proof. Suppose for the sake of contradiction that F0 is a universal graph for . Let F0 be F 0 obtained from F0 by adding a new vertex v adjacent to all vertices in F0. We claim that F0 is in . Suppose for the sake of contradiction that F 0 contains a subgraph G isomorphic F 0 to a graph in . Since F does not contain G, it must be that v V (G). By assumption, G 0 ∈ G v contains a graph in , and that graph is also a subgraph of F0. This contradiction − 0 G shows that F0 . Since F0 is universal for , F0 contains a subgraph F1 isomorphic to 0 ∈ F F F0. Let v1 be the vertex in F1 corresponding to v. Then F1 v1 contains a subgraph F2 0 − isomorphic to F0. Let v2 be the vertex in F2 corresponding to v. We repeat this argument and obtain a sequence v, v1, v2,... of vertices in F0 inducing Kℵ0 . Thus F0 contains every graph, which contradicts the assumption that = . G 6 ∅

Numerous graph classes satisfy the assumptions in Lemma 3.14 for an appropriately deﬁned class . In particular, the following graph classes have no universal element: G

• chordal graphs containing no Kℵ0 ;

• graphs with no Kℵ0 minor (which was proved in [56]);

• graphs containing no Kℵ0 subdivision (which was left open in [56]); • graphs containing no subdivision of the d-regular tree; • graphs containing no tree where all the vertices, except possibly one, have degree d; • graphs in which each subgraph has a vertex of finite degree; • graphs that do not contain infinitely many pairwise disjoint one-way infinite paths; • graphs that do not contain d pairwise disjoint one-way infinite paths (for any fixed d N). ∈

4 Universality for Trees and Treewidth

This section proves universality results for trees and graphs of given treewidth. While such results are well known, our construction is novel and leads to strengthenings in terms of orientation- and labelling-preserving isomorphisms, which are a key tool used in our constructions for graphs deﬁned by a tree-decomposition (Section 4.3), for Kt-minor-free graphs (Section 6.2), and for locally ﬁnite graphs (Section 6.3).

32 4.1 Universal Trees

The following universal graph for (countable) trees is folklore. Let be the graph with T

V ( ) := (x1, . . . , xn): n N0, x1, . . . , xn N and T { ∈ ∈ } E( ) := (x1, . . . , xn)(x1, . . . , xn, xn+1): n N0, x1, . . . , xn, xn+1 N . T { ∈ ∈ } Theorem 4.1. is universal for the class of trees. T

Proof. Consider to be rooted at vertex (). From each vertex (x , . . . , x ) in there is a T 1 n T unique path (x , . . . , x ), (x , . . . , x − ), (x , . . . , x − ),..., (x ), () to the root. Thus is 1 n 1 n 1 1 n 1 1 T a tree. Since V ( ) is a countable union of countable sets, is countable. We now show T T that every countable tree X is isomorphic to a subtree of . Root X at an arbitrary vertex T r. Let ` : V (X) N be an injective function, which exists since X is countable. For → each non-root vertex w of X, if (r, w1, w2, . . . , wn) is the path from r to w in X, then let φ(w) := (`(w1), `(w2), . . . , `(wn)). Finally, let φ(r) := (). Then φ is an isomorphism from X to a subtree of . T

A labelling of a graph G is a function f : V (G) E(G) N. We now prove a strengthening ∪ → of Theorem 4.1, which will be used in Section 4.3. Lemma 4.2. There is a labelling of such that for every 1-orientation of , for every T T rooted tree T , and for every labelling of T there is a label-preserving orientation-preserving isomorphism from T to a subtree of . T

Proof. Since every vertex of has infinite degree, there is a labelling of such that for T T every vertex v of and for all α, β N there are infinitely many neighbours w of v for which T ∈ w is labelled α and vw is labelled β. Fix any 1-orientation of . Now, for every vertex v of T and for all α, β N there are infinitely many children w of v for which w is labelled α T ∈ and vw is labelled β.

Let T be a tree rooted at r, and let f be any labelling of T . We now construct a label- preserving orientation-preserving isomorphism φ from T to a subtree of . Let φ(r) be any T vertex in labelled f(r). Consider the vertices of T in order of non-decreasing distance T from r. Let v be a vertex of T such that φ(v) is already defined, but φ(w) is undefined for every child w of v. Let W be the set of children of v in T . For all α, β N there are ∈ infinitely many children z of φ(v) for which z is labelled α and vz is labelled β. So φ can be extended to W so that for each w W , we have f(w) equals the label assigned to φ(w) ∈ in , and f(vw) equals the label assigned to φ(vw) in . Hence f is a label-preserving T T orientation-preserving isomorphism from T to a subtree of . T

As an aside, we show that Lemma 4.2 is not true for (unrooted) 1-oriented trees T . Suppose that there is a labelling and 1-orientation of the universal tree such that for every labelled T and 1-oriented tree T , there is a label-preserving orientation-preserving isomorphism from T to a subtree of . In particular, for every labelled anti-directed 1-way inﬁnite path P , there T is a label-preserving orientation-preserving isomorphism from P to . There are uncountably T many labelled anti-directed 1-way inﬁnite paths (since at each vertex there are at least two

33 choices of label). However, for each vertex v of there is exactly one anti-directed 1-way T infinite path starting at v. Hence some labelling of the anti-directed 1-way infinite path does not appear in . T To complete this subsection, we define a universal tree of given maximum degree. Let (d) T be the graph with

(d) V ( ) := (x1, . . . , xn): n N0, x1, . . . , xn [d], i [n 1] xi = xi+1 T { ∈ ∈ ∀ ∈ − 6 } E( (d)) := (x , . . . , x )(x , . . . , x , x ): T { 1 n 1 n n+1 n N0, x1, . . . , xn, xn+1 [d], i [n] xi = xi+1 . ∈ ∈ ∀ ∈ 6 } A proof analogous to that of Theorem 4.1 shows that (d) is a tree, and by definition, (d) T T is d-regular. Indeed, up to isomorphism, (d) is the unique d-regular tree, sometimes called T the Bethe lattice or infinite Cayley tree [18, 132]. This tree is used in Section 4.4 as the basis for the definition of a universal graph of given simple treewidth.

4.2 Universality for Treewidth

Chordal graphs and the theory of simplicial decompositions [53, 94–96, 159] can be used to define universal treewidth-k graphs. We take a somewhat different approach that gives a new and explicit definition of a universal treewidth-k graph . This approach allows us Tk to derive further properties of , regarding label- and orientation-preserving isomorphisms Tk (Lemma 4.6), that are essential for results in Section 6.2.

The following deﬁnitions lead to a new characterisation of graphs with given treewidth. Let c be a colouring of a 1-oriented tree T . Let T c be the graph with vertex set V (T c ) = V (T ), h i h i where vw E(T c ) if and only if v is an ancestor of w in T and v is the only vertex on the ∈ h i vw-path in T coloured c(v). Where it is clear from the context, we implicitly consider the edges vw of T c to be oriented from the ancestor v to the descendent w. h i Lemma 4.3. For every 1-oriented tree T and (k + 1)-colouring c of T , the graph T c h i is chordal with no Kk+2 subgraph, and is simplicially k-oriented. Conversely, let G be a connected chordal graph with no K subgraph. Fix a k-orientation of G. Then G T c k+2 ⊆ h i for some 1-oriented spanning tree T of G and for every (k + 1)-colouring c of G.

Proof. We ﬁrst prove that T c is chordal with no K subgraph. By construction, each h i k+2 vertex w has in-degree at most k in T c , and the in-neighbourhood of v in T c lies on a h i h i single directed path in T that ends at v. Consider two directed edges uw and vw in T c . h i So both u and v are ancestors of w in T , and c(u) = c(v). Without loss of generality, u is 6 an ancestor of v. Since uw is an edge, u is the only vertex on the uw-path in T coloured c(u). In particular, u is the only vertex on the uv-path in T coloured c(u). So uv is an edge of T c . Hence the in-neighbourhood of w in T c is a clique, and T c is simplicially h i h i h i k-oriented. By Lemma 2.7, T c is chordal. Each vertex in T c has in-degree at most k. h i h i Thus T c contains no K . h i k+2 Let Z be the set of oriented edges vw of G for which there is no vertex x such that vxw is an oriented path in G. Let T be the spanning subgraph of G with edge-set Z (ignoring

34 the edge orientation). We claim that T is a spanning tree of G with in-degree at most 1 at every vertex. Suppose that some vertex v has in-degree at least 2 in T . Let xv and yv be two incoming edges incident to v in T . Since the in-neighbourhood of v is a clique in G, without loss of generality, xy is an oriented edge in G. Thus xyv is an oriented path in G, implying that xv is not in Z. This contradiction shows that T has in-degree at most 1. Since an acyclically oriented cycle contains a vertex with in-degree 2, T contains no cycle. We now show that T is connected. For any two vertices a and b, since G is connected, there is an ab-path in G (ignoring the edge orientation). For each oriented edge vw in P , if vw is not in Z, then there is a vertex x such that vxw is an oriented path in G. Replace vw by vx and xw in P . Repeat this operation until every edge of P is in Z. We obtain an ab-path in T (ignoring the edge orientation). Thus T is connected. Hence T is a spanning tree of G with in-degree at most 1 at each vertex.

Suppose for the sake of contradiction that for some oriented edge vw of G, there is no directed vw-path in T . Let vw be such an edge that minimises the distance between v and w in T (ignoring edge orientations). Let P be the vw-path in T (ignoring orientations). Let x be the least common ancestor of v and w in T . So P is oriented from x to v and from x to w (and x = v and x = w). Let y be the neighbour of w on the vw-path in T . Since the 6 6 in-neighbourhood of w is a clique, vy or yv is an edge of G, which contradicts the choice of vw. Thus for every oriented edge vw of G, there is a directed vw-path in T .

Consider any oriented path v1, v2, . . . , vp in G, where v1vp is an edge of G. Since the in- neighbourhood of vp is a clique, v1vp−1 is an edge of G. Since the in-neighbourhood of vp−1 is a clique, v1vp−2 is an edge of G. Repeating this argument, v1vi is an edge of G, for each i [2, p]. ∈ Let c be any (k + 1)-colouring of G, which exists since chordal graphs are perfect and by the de Bruijn–Erdős Theorem [49]. Consider an oriented edge vw of G. As shown above there is a directed path P from v to w in T , and v is adjacent to every vertex in P . Thus c(v) = c(x) for every vertex x in P . By construction, vw E(T c ). Hence G T c . 6 ∈ h i ⊆ h i Lemma 4.4. The following are equivalent for a graph G and k N: ∈ (a) G has treewidth at most k, (b) there is a 1-oriented tree T and a (k + 1)-colouring c of T such that V (T ) = V (G) and G T c , ⊆ h i (c) G is a spanning subgraph of a chordal graph with no Kk+2 subgraph.

Proof. (a) = (b): Let G be a graph with treewidth at most k. Apply Lemma 2.10 to ⇒ obtain a rooted tree T , a tree-decomposition (B : w V (T )) of G, and a (k + 1)-colouring w ∈ c of G. We now show that G T c . By construction, V (G) = V (T c ). Consider an edge ⊆ h i h i vw E(G). Then c(v) = c(w). Without loss of generality, w is a descendant of v. Consider ∈ 6 some vertex x on the vw-path in T . Since v B B , it follows that v is in B , implying ∈ v ∩ w x c(v) = c(x) by Lemma 2.10. Hence vw E(T c ) by the deﬁnition of E(T c ). Therefore 6 ∈ h i h i G T c . ⊆ h i (b) = (c): Assume T is a 1-oriented tree and c is a (k + 1)-colouring of T such that ⇒ V (T ) = V (G) and G T c . By Lemma 4.3, T c is a chordal graph with no K ⊆ h i h i k+2

35 subgraph containing G as a spanning subgraph, as desired.

(c) = (a): Assume G is a spanning subgraph of a chordal graph G0 with no K subgraph. ⇒ k+2 By Lemma 2.7, G0 has a tree-decomposition in which each bag is a clique, and therefore of size at most k + 1. The same tree-decomposition is a tree-decomposition of G. So tw(G) 6 k.

We now define a universal treewidth-k graph. Let be the universal tree defined in T Theorem 4.1 rooted at vertex (). For k N, let k := c , where c is a colouring of with ∈ T T h i T colour-set [0, k], where every vertex coloured i [0, k] has infinitely many children of each ∈ colour in [0, k] i . \{ } Theorem 4.5. is universal for the class of graphs with treewidth at most k. Tk

Proof. c has treewidth at most k by Lemma 4.4 and since c uses k +1 colours. Conversely, T h i let G be a graph with treewidth at most k. By Lemma 4.4, there is a tree T rooted at some vertex r, and there is a (k + 1)-colouring α : V (T ) [0, k] such that V (T ) = V (G) and → G T α . By Lemma 4.2 there is a colour-preserving orientation-preserving isomorphism ⊆ h i from T (coloured by α) to a subtree of (coloured by c). Thus T α is isomorphic to a T h i subgraph of c . Hence G is isomorphic to a subgraph of c . T h i T h i

We now prove two strengthenings of Theorem 4.5 that are used in Section 6.2. The ﬁrst is an analogue of Lemma 4.2.

Lemma 4.6. For each k N, there is a labelling and a rooted k-orientation of k such that ∈ T for every chordal graph G with no Kk+2 subgraph, for every labelling of G and for every rooted k-orientation of G, there is a label-preserving orientation-preserving isomorphism from G to a subgraph of . Tk

Proof. Recall that is the universal tree rooted at vertex (), and that is the universal T Tk treewidth-k graph with V ( ) = V ( ). We may orient the edges of E( ) such that for Tk T Tk each oriented edge vw, v is an ancestor of w in , and each vertex of has in-degree at T Tk most k. By definition, has a (k + 1)-colouring such that each vertex v of has infinitely Tk T many children of each colour (distinct from the colour assigned to v). Label each vertex of such that for each vertex v of , for each colour α distinct from the colour assigned Tk T to v, and for each β N, there are infinitely many children of v coloured α and labelled β. ∈ This is possible since v has infinitely many children coloured α. Consider each vertex w of . Let v w, . . . , v w be the edges of incoming to w. So p k. Let W be the siblings of Tk 1 p Tk 6 w assigned the same colour and the same label as w. So W is infinite. Since every vertex x W has the same colour as w and the same parent as w in , by the construction of k, ∈ T p T we have that v1x, . . . , vpx are the edges of k incoming to x. Since N is countable, there is T p a labelling of the edges vix (where i [p] and x W ), such that for each (γ1, . . . , γp) N ∈ ∈ ∈ there are infinitely many vertices x W , such that the edge v x of is labelled γ for each ∈ i Tk i i [p]. Call this property (?). ∈ Now, let G be a chordal graph with no Kk+2 subgraph, with a fixed k-orientation and labelling f. Our goal is to show that there is a label-preserving orientation-preserving isomorphism from G to a subgraph of . Since contains infinitely many pairwise disjoint Tk Tk

36 subgraphs isomorphic to itself, we may assume that G is connected. Let r be the root of G. By Lemma 4.3, G is an orientation-preserved subgraph of T c for some 1-oriented h i spanning tree T of G and for every (k + 1)-colouring c of G. We now define an isomorphism φ from G to a subgraph of . We define φ(v) in order of non-decreasing dist (r, v). Let Tk T φ(r) be any vertex of labelled f(r). Let v be a vertex of T for which φ(v) is already Tk defined, but φ is defined for no child of v. For α [0, k] and β N, let Wα,β be the set of ∈ ∈ children w of v in T , such that c(w) = α and f(w) = β. Consider such a w W . Let ∈ α,β v w, . . . , v w be the edges of T c incoming to w. So p k. By the choice of v, we have 1 p h i 6 that φ(v ), . . . , φ(v ) are already defined. Since every vertex x W has the same colour 1 p ∈ α,β as w and the same parent as w in T , by the construction of T c , we have that v1x, . . . , vpx p h i are the edges of T c incoming to x. For (γ1, . . . , γp) N , let Wα,β,γ1,...,γp be the set of h i ∈ vertices x W such that f(v x) = γ for each i [p]. By property (?), there are infinitely ∈ α,β i i ∈ many children z of φ(v) in , such that z is coloured α and labelled β, and φ(v )z is labelled T i γ for each i [p]. Injectively map W to these vertices z under φ. Hence φ is a i ∈ α,β,γ1,...,γp label-preserving orientation-preserving isomorphism from T c to a subgraph of . Since G h i Tk is an orientation-preserved subgraph of T c , the result follows. h i

Lemma 4.7. Let G be a chordal graph with no Kk+2 subgraph. Fix a rooted k-orientation of G. Let Q be a chordal graph containing G as a spanning subgraph, with no Kk+2 subgraph, with a rooted k-orientation that preserves the given k-orientation of G, and subject to these conditions is edge-maximal. Then for every labelling of Q, there is an orientation-preserving label-preserving isomorphism from Q to an induced subgraph of . Tk

Proof. By Lemma 4.6, there is a labelling and rooted k-orientation of such that for every Tk rooted k-orientation of Q, there is a label-preserving orientation-preserving isomorphism φ from Q to a subgraph of . Let Q0 be the graph with vertex-set V (Q), where vw E(Q0) Tk ∈ whenever φ(v)φ(w) E( ). Orient Q0 by the corresponding orientation of , which ∈ Tk Tk preserves the orientation of G, since Q preserves the orientation of G and φ preserves the orientation of Q. Since Q0 is isomorphic to an induced subgraph of (which is chordal with Tk no K subgraph), Q0 is chordal with no K subgraph. By construction, Q Q0. By the k+2 k+2 ⊆ maximality of Q, we have Q = Q0. So φ(Q) is an induced subgraph of . Tk

4.3 Graphs Deﬁned by a Tree-Decomposition

Recall that (U) is the class of graphs that have a tree-decomposition over the graph U. D We now construct a universal graph for (U). This result is used to construct a universal D graph for graphs of given simple treewidth (Section 4.4), as well as to construct a graph that contains every X-minor-free graph (Section 5.5). This is the ﬁrst place where we see the utility of our results about orientation and label-preserving isomorphisms developed above.

Lemma 4.8. For every graph U containing no Kℵ0 subgraph, there is a universal graph Ub for (U). D

Proof. Let be the set of all pairs ((v , . . . , v ), (w , . . . , w )), where v , . . . , v and C 1 k 1 k { 1 k} w1, . . . , wk are cliques in U for any k N. Since U is countable with no Kℵ0 subgraph, { } ∈ C is countable. Enumerate = C ,C ,... . Let be the universal tree 1-oriented and C { 1 2 } T

37 labelled, such that every vertex is labelled 1, and for every vertex v of and every i N T ∈ there are inﬁnitely many children w of v with vw labelled i. Let Ub be the graph obtained from as follows. First, for each vertex v of , introduce a copy Uv of U in Ub, where Uv T T and U are disjoint for all distinct v, w V ( ). Second, for each edge vw E( ), if vw is w ∈ T ∈ T labelled i N and Ci = ((v1, . . . , vk), (w1, . . . , wk)), then for each j [k], identify vertex vj ∈ ∈ in Uv with vertex wj in Uw. This deﬁnes Ub.

For each vertex v V ( ), let Ubv be the subgraph of Ub corresponding to the copy Uv (after ∈ T the above vertex identiﬁcations). Then (V (Ubv): v V ( )) is a tree-decomposition of Ub, ∈ T where each torso is isomorphic to U. Thus Ub is in (U). D We now show that (U) contains every graph G in (U). So G has a tree-decomposition D D (B : x V (T )), such that for each node x V (T ), there is an isomorphism φ from the x ∈ ∈ x torso of x to a subgraph of U. Root T at an arbitrary node r. Label each vertex of T by 1. Label each oriented edge xy of T as follows. Say B B = z , . . . , z , which is a x ∩ y { 1 k} clique in the torsos of x and y. For j [k], let v and w be the vertices of U such that ∈ j j f (z ) = v and f (z ) = w . So v , . . . , v and w , . . . , w are cliques of U. Label vw x j j y j j { 1 k} { 1 k} by i N so that Ci = ((v1, . . . , vk), (w1, . . . , wk)). By Lemma 4.2, there is a label-preserving ∈ orientation-preserving isomorphism φ from T to . By construction, S Ub contains T v∈V (T ) φ(v) a subgraph isomorphic to G.

Lemma 4.8 can be extended as follows to allow for tree-decompositions with adhesion k. The proof is identical to that Lemma 4.8, except that in the deﬁnition of we only consider C cliques with size at most k.

Lemma 4.9. For every graph U and k N, there is a universal graph Ubk for the class ∈ (U). Dk

4.4 Simple Treewidth

A tree-decomposition (Bx : x V (T )) of a graph G is k-simple, for some k N, if it has ∈ ∈ width at most k, and for every set S of k vertices in G, we have x V (T ): S B 2. |{ ∈ ⊆ x}| 6 The simple treewidth of a graph G, denoted by stw(G), is the minimum k N such that G has ∈ a k-simple tree-decomposition. Simple treewidth appears in several places in the literature under various guises [105, 115, 124, 170]. The following facts are well known: A connected finite graph has simple treewidth 1 if and only if it is a path. A finite graph has simple treewidth at most 2 if and only if it is a outerplanar. A finite graph has simple treewidth at most 3 if and only if it has treewidth 3 and is planar [115]. The edge-maximal finite graphs with simple treewidth 3 are ubiquitous objects, called planar 3-trees in structural graph theory and graph drawing [14, 115], called stacked polytopes in polytope theory [38], and called Apollonian networks in enumerative and random graph theory [78]. It is also known and easily proved that tw(G) 6 stw(G) 6 tw(G) + 1 for every finite graph G (see [105, 170]). A similar proof shows this result for infinite graphs.

Simple treewidth can be characterised in terms of chordal supergraphs. The following lemma for ﬁnite graphs is due to Wulf [170]. Let W be the graph with vertex set u, v, w, x , . . . , x , k { 1 k} where each of u, x , . . . , x , v, x , . . . , x and w, x , . . . , x are cliques. { 1 k} { 1 k} { 1 k}

38 Lemma 4.10. A graph G has simple treewidth at most k N if and only if G is a subgraph ∈ of a chordal graph containing no Kk+2 or Wk subgraph,

Proof. By Lemma 2.8, in every tree-decomposition of Wk with width k, the cliques u, x , . . . , x , v, x , . . . , x and w, x , . . . , x are distinct bags. Therefore, x , . . . , x { 1 k} { 1 k} { 1 k} { 1 k} is contained in at least three bags, and such a tree-decomposition is not k-simple. Thus stw(Wk) > k. Now consider a graph G with simple treewidth at most k. Starting from a k-simple tree- decomposition of G, let G0 be the graph obtained from G by adding an edge between any two non-adjacent vertices in a common bag. We obtain a k-simple tree-decomposition of G0, implying stw(G0) k. By Lemma 2.7, G0 is chordal, and W G0 since stw(W ) > k. By 6 k 6⊆ k Lemma 2.8, K G0. k+2 6⊆ We now prove that if G is a subgraph of a chordal graph with no Kk+2 or Wk subgraph, then stw(G) 6 k. It suﬃces to prove the result when G is chordal with no Kk+2 or Wk subgraph. Let X : i I be the maximal cliques of G. Let T be the graph with vertex-set { i ∈ } I, where ij E(T ) if and only if X X = and i = j. Since G is chordal, T is a tree and ∈ i ∩ j 6 ∅ 6 (X : i V (T )) is a tree-decomposition of G. Since K G, the width is at most k. If i ∈ k+2 6⊆ some set X of k vertices appears in distinct bags X ,X ,X , then X X X induce W . a b c a ∪ b ∪ c k Thus (X : i V (T )) is a k-simple tree-decomposition of G, and stw(G) k. i ∈ 6

(k+1) For k N, define the directed graph k as follows. Recall that is the universal ∈ R T (k + 1)-regular tree defined in Section 4.1. Fix an orientation of (k+1) in which each vertex T has in-degree 1 and out-degree k, which exists by Lemma 2.4. Let c : V ( (k+1)) [0, k] be T → a colouring of V ( (k+1)) such that for each vertex x with out-neighbours y , . . . , y , all of T 1 k x, y1, . . . , yk are assigned different colours. This is possible since, in the graph obtained from (k+1) by adding a k-clique on the out-neighbours of each vertex, each biconnected subgraph T is Kk+1 (since by the de Bruijn–Erdős Theorem [49] and considering the cut-block-tree, if every biconnected subgraph of a graph G is c-colourable, then so is G)). Now define := T c . For example, Figures4 and5 illustrate and . Rk h i R2 R3 Lemma 4.11. has simple treewidth k. Rk

Proof. Let (k+1) be the naturally oriented tree and c be the colouring used in the deﬁnition T of . Note that c is a (k + 1)-colouring of . For each vertex x of (k+1), let B be the Rk Rk T x closed in-neighbourhood of x.

We now show that (B : x V ( (k+1))) is a k-simple tree-decomposition of . By x ∈ T Rk construction, each edge vw of has its endpoints in B . In fact, B is a clique of size Rk w w k + 1 in , and c(x) = c(y) for all distinct x, y B . Consider the set of bags that contain Rk 6 ∈ w a given vertex u. Say u is in the bag B for some w = u. Then uw E( ) and w is a w 6 ∈ Rk descendant of u. Let v be any vertex on the uw-path in (k+1). Since uw E( ), we have T ∈ Rk uv E( ) and u is also in B . Hence the sets of bags that contain u form a connected ∈ Rk v subtree of (k+1). Thus (B : u V ( (k+1))) is a tree-decomposition of of width at T u ∈ T Rk most k.

39 We now show that this tree-decomposition is k-simple. Since every bag is a clique, it suﬃces to show that every set S of k vertices is contained in at most two bags. Let C be the set of colours assigned to vertices in S. Since S is a clique, C = k. Let α be the element of | | [0, k] C. \ Every edge in S is between an ancestor and a descendant in (k+1). Let u = (x , . . . , x ) be T 1 n the vertex in S that is furthest from the root in (k+1). By construction, S B . Consider T ⊆ u a vertex v = u such that S B . For each vertex w S u , we have u B . Thus 6 ⊆ v ∈ \{ } 6∈ w v S. Since B is a clique, v is coloured α. Since S B , v is a descendant of u in (k+1). 6∈ v ⊆ v T Suppose that v is not a child of u in T . Let w be any internal vertex on the uv-path in T . Then the colour of w is assigned to some vertex in S, which implies that S B . Thus v is 6⊆ v a child of u. Hence v = (x1, . . . , xn, α), and Bu and Bv are the only bags containing S. Therefore (B : u V ( (k+1))) is a k-simple tree-decomposition of . Since contains u ∈ T Rk Rk Kk+1, it has simple treewidth k.

There are infinite graphs with simple treewidth k that are not subgraphs of . The first Rk example is the disjoint union of countably many infinite 2-way paths, which has simple treewidth 1, but is not a subgraph of . Examples for all k are easily constructed. R1 Nevertheless we have the following characterisation of graphs with simple treewidth k.

Lemma 4.12. For k N, a graph has simple treewidth k if and only if it has a tree- ∈ decomposition over with adhesion k 1. Rk −

Figure 4: Illustration of the outerplanar graph with the 1-oriented 3-regular universal R2 tree highlighted.

40 Figure 5: Illustration of the planar graph with the 1-oriented 4-regular universal tree R3 highlighted.

Proof. Let G be a graph with simple treewidth k. Let (B : x V (T )) be a k-simple x ∈ tree-decomposition of G. We may assume that B is a clique in G for each node x V (T ), x ∈ and that B B and B B for each edge xy E(T ). x 6⊆ y y 6⊆ x ∈ Say an edge xy E(T ) is thick if B B = k (which implies B = B = k + 1). For ∈ | x ∩ y| | x| | y| each vertex x V (T ) there are at most k + 1 possible values for B B where xy is a thick ∈ x ∩ y edge incident to x. So if x is incident with k + 2 thick edges, then B B = B B for | x ∩ y| | x ∩ z| distinct y, z N (x), implying three bags contain B B , which contradicts the k-simplicity ∈ T x ∩ y of the tree-decomposition. Hence each vertex x V (T ) is incident with at most k + 1 thick ∈ edges. Let F be the forest with V (F ) := V (T ) consisting only of the thick edges. Hence F has maximum degree at most k + 1. S Let F1,F2,... be the connected components of F . Let Gi := G[ Bx : x V (Fi)] for i N. { ∈ ∈ For each vertex v V (G ), initialise F to be the subtree F [ x V (F ): v B ]. Let ∈ i i,v i { ∈ i ∈ x} c : V (G) [0, k] be a (k + 1)-colouring of V (G) such that vertices in a common bag are → assigned distinct colours. We now show that G F c (after a small change to F in one i ⊆ ih i i case). Recall that Fi has maximum degree at most k + 1.

41 First suppose that Fi has a vertex s of degree at most k. Orient every edge of Fi away from s. Now F is 1-oriented, and every node of F has out-degree at most k. Say B = v , . . . , v i i s { 1 p} where p [k]. Add new vertices x , . . . , s − to F , where (s , . . . , s − , s) is a directed path, ∈ 1 p 1 i 1 p 1 and let B := v , . . . , v for each i [p 1]. Observe that B is still a clique for each si { 1 i} ∈ − x node x V (F ), and vertices in a common bag are still assigned distinct colours under c. ∈ Consider Fi to be rooted at s1. Now every node of Fi has out-degree at most k and in-degree 1, except for s which has in-degree 0. Note that (a) B = 1, and (b) for every non-root 1 | s1 | node x of F , if yx is the incoming arc at x, then B B = 1. In case (a) rename s by i | x \ y| 1 the element of B . In case (b), rename x by the element of B B . After this renaming, s1 x \ y V (F ) = V (G ). By the argument in the proof of Lemma 4.4(b), G F c . i i i ⊆ ih i Now assume that Fi is (k + 1)-regular. Assign each edge xy of Fi the unique colour not present on the vertices of B B , which is well-defined since there are exactly k vertices in x ∩ y B B and they are assigned distinct colours. Observe that F is now properly (k + 1)-edge x ∩ y i coloured, and every colour is present at each node of Fi. Let P = (. . . , v−2, v−1, v0, v1, v2,... ) be an infinite 2-way path in Fi, where each edge vjvj+1 is coloured j mod (k + 1). Orient each edge v v of P from v to v and orient each edge of F E(P ) away from P . Note j j+1 j j+1 i − that each vertex of Fi has in-degree 1 and out-degree k. For each vertex v of Gi coloured α [0, k], by construction, each edge xy of F is not coloured α. Since F intersects P ∈ i,v i,v in a subpath, Fi,v intersects at most k edges in P . In particular, some edge of P is not in Fi,v. If Fi,v does not intersect P , then the vertex of Fi,v closest to P has in-degree 0 in Fi,v. If F does intersect P , then the vertex v in P F with a minimum has in-degree 0 in i,v a ∩ i,v Fi,v since va−1va is the incoming edge at va and va−1 is not in Fi,v We have shown that Fi,v has a vertex x of in-degree 0 for each vertex v V (G ). Suppose that x = x for distinct v ∈ i v w vertices v, w V (G ). Let x := x . Let yx be the incoming arc at x. Since x has in-degree ∈ i v 0 in both F and F , we have v, w B B , which contradicts the fact that xy is thick. i,v i,w ∈ x ∩ y Hence x = x for distinct vertices v, w V (G ). Rename x by v. Now V (F ) = V (G ). v 6 w ∈ i v i i By the argument in the proof of Lemma 4.4(b), G F c . i ⊆ ih i Note that up to isomorphism (k+1) is the unique oriented tree with in-degree 1 and out- T degree k at every node. So F is a subtree of (k+1). By the definition of , G is isomorphic i T Rk i to a subgraph of . Let T 0 be obtained from T by contracting each subtree F into a vertex Rk i x . Then (V (G ): x V (T 0)) is a tree-decomposition of G. Since each bag B is a clique in i i i ∈ x G, the torso of each node x is G itself. Hence (V (G ): x V (T 0)) is a tree-decomposition i i i i ∈ of G over . Rk We now prove the converse. Let (B : x V (T )) be a tree-decomposition of a graph G x ∈ over with adhesion k 1. Let G be the torso of x V (T ). So G is isomorphic to a Rk − x ∈ x subgraph of . By Lemma 4.11, G has a k-simple tree-decomposition (Bx : y V (T x)). Rk x y ∈ Initialise F to be the disjoint union of T x : x V (T ) , Associate with each node y of F the { ∈ } corresponding bag B := Bx. Then for each edge x x V (T ), let Bx1 and Bx2 be bags y y 1 2 ∈ y1 y2 respectively in the tree-decomposition of G and G with B B Bx1 Bx2 , which x1 x2 x1 ∩ x2 ⊆ y1 ∩ y2 exist since B B is a clique in G and in G (by the definition of torso). Add an edge x1 ∩ x2 x1 x2 in F between y and y . We obtain a tree-decomposition (B : y V (F )) of G. Since the 1 2 y ∈ tree-decomposition of each G is k-simple, and B B k 1 for each edge x x of T , x | x1 ∩ x2 | 6 − 1 2 the tree-decomposition (B : y V (F )) of G is also k-simple. Hence G has simple treewidth y ∈ at most k.

42 k−1 Let k := ck deﬁned in Lemma 4.9. Then Lemmas 4.9 and 4.12 imply: S R

Theorem 4.13. For each k N, the graph k is universal for the class of graphs with simple ∈ S treewidth k.

We now focus on graphs with simple treewidth 2 or 3.

Theorem 4.14. is universal for the class of outerplanar graphs. S2

Proof. If a graph G has a tree-decomposition of adhesion 1 in which every torso is outerplanar, then G is also outerplanar. If a graph G has a tree-decomposition of adhesion 1 in which every torso has treewidth at most 2, then G also has treewidth at most 2. Now has S2 a tree-decomposition of adhesion 1 in which every torso is isomorphic to (which is R1 outerplanar). Thus is outerplanar. S2 We now prove the converse. That is, contains every outerplanar graph. By Theorem 4.13, S2 it suffices to show that every outerplanar graph has simple treewidth 2. Since simple treewidth is extendable (Lemma 4.18 below), it suffices to show that every finite outerplanar graph has simple treewidth 2, which is a folklore and easily proved result [105, 124, 170].

Theorem 4.15. is universal for the class of planar graphs with treewidth 3. S3

Proof. If a graph G has a tree-decomposition of adhesion 2 in which every torso is planar, then G is also planar. If a graph G has a tree-decomposition of adhesion 2 in which every torso has treewidth at most 3, then G also has treewidth at most 3. Now has a tree- S3 decomposition of adhesion 2 in which every torso is isomorphic to (which is planar with R3 treewidth 3). Thus is planar with treewidth 3. S3 We now prove the converse. That is, contains every planar graph with treewidth 3. S3 By Theorem 4.13, it suffices to show that every planar graph with treewidth 3 has simple treewidth 3. Since simple treewidth is extendable (Lemma 4.18 below), it suffices to show that every finite planar graph with treewidth 3 has simple treewidth 3. This was proved by Knauer and Ueckerdt [105] (also see [170]) using the result of Kratochvíl and Vaner [115], who showed that for every finite planar graph G of treewidth 3 there is a 3-tree (an edge-maximal graph of treewidth 3) that is planar and contains G as a spanning subgraph.

4.5 Treewidth Extendability

Recall that a graph class Γ is extendable if the following property holds for every graph G: if every ﬁnite subgraph of G is in Γ, then G is in Γ. (Recall that graphs are assumed to be countable.)

Thomas [157] proved the following result.

Lemma 4.16 ([157]). Treewidth is extendable.

See [121, 160] for simpler proofs of Lemma 4.16. The proof of Lemma 4.16 due to Thomassen [160] generalises as follows. Deﬁne a forbidden pattern to be a sequence

43 = ( k : k N) where k is any class of finite graphs, such that for every k N, F F ∈ F ∈ every graph in has a subgraph in . Define the -width of a graph G to be the Fk+1 Fk F minimum k N such that G is a subgraph of a chordal graph containing no subgraph in ∈ . If there is no such k, then G has infinite -width. Observe that if = K then Fk F Fk { k+2} -width equals treewidth by Lemma 4.4, and if = K ,W then -width equals F Fk { k+2 k} F simple treewidth by Lemma 4.10. Lemma 4.17. -width is extendable for every forbidden pattern . F F

Proof. Let G be an infinite graph such that every finite subgraph of G has -width at most F k. Our goal is to show that G has -width at most k. Let be the class of graphs G0 such F G that V (G0) = V (G), E(G) E(G0) and every finite subgraph of G0 has -width at most ⊆ F k. Thus G . Consider the partial order defined on by inclusion (of the corresponding ∈ G G edge sets). 00 S 0 00 00 Let be a chain in . Let G := 0 G . We claim that G . Certainly, V (G ) = V (G) X G G ∈X ∈ G and E(G) E(G00). If G00 contains a finite subgraph H with -width greater than k, then ⊆ F since is totally ordered by inclusion and H is finite, some graph G000 in would contain H, X X implying that the -width of G000 would be greater than k, which contradicts the definition F of . Hence G00 is an upper bound on in . G X G By Lemma 2.3, contains a maximal element G ; that is, E(G0) E(G ) for every G0 . G 0 ⊆ 0 ∈ G We claim that G0 is chordal. Suppose on the contrary that G0 has a chordless cycle C of length at least 4. For each pair of non-adjacent vertices x, y V (C), let G be the graph ∈ xy obtained from G0 by adding the edge xy. Since G0 is maximal with respect to inclusion, G . Since V (G ) = V (G) and E(G) E(G ), it must be that G contains a finite xy 6∈ G xy ⊆ xy xy subgraph H with -width greater than k. Let H be the union of H xy taken over every xy F xy − pair of non-adjacent vertices x, y V (C). Since C is finite and each H is finite, H is ∈ | | xy finite. By construction, H is a subgraph of G , which is in . Thus H has -width at most k. 0 G F By the definition of -width, H is a subgraph of a chordal graph H0 containing no element F of . Since H0 is chordal, it contains some chord, say xy, of C. Now H H H0. Since Fk xy ⊆ ⊆ H0 has -width at most k, so does H , which is a contradiction. Hence G is chordal. F xy 0 Since G , every finite subgraph of G has -width at most k. In particular, no element 0 ∈ G 0 F of is a subgraph of G (since elements of have -width greater than k). Hence G Fk 0 Fk F 0 has -width at most k. Therefore G has -width at most k since G G . F F ⊆ 0

Lemma 4.17 implies Lemma 4.16 as well as the following. Lemma 4.18. Simple treewidth is extendable.

The following result generalises Lemma 4.16, and is a direct corollary of a result of Kříž and Thomas [119, Theorem 3.9]. Lemma 4.19 ([119]). For every hereditary and extendable class of graphs Γ, the class (Γ) D is extendable.

For a graph G and c N0, let twc(G) be the minimum integer k such that tw(G S) k ∈ − 6 for some S V (G) with S c. Of course, tw(G) = tw (G) and tw(G) tw (G) + c. We ⊆ | | 6 0 6 c need the following generalisation of Lemma 4.16.

44 Lemma 4.20. twc is extendable for every c N0. ∈

Lemma 4.20 follows from the next lemma by induction on c (with Lemma 4.16 in the base case). Lemma 4.21. Let Γ be a monotone and extendable graph class. Let Γ+ be the class of graphs G such that G v is in Γ for some vertex v of G. Then Γ+ is monotone and extendable. −

Proof. Since Γ is monotone, so too is Γ+. It remains to show that Γ+ is extendable. Let + G be a graph such that every ﬁnite subgraph of G is in Γ . Let v1, v2,... be an arbitrary ordering of V (G). For i N, initialise ∈

X := (i, j): j [i],G[ v , . . . , v − , v , . . . , v ] Γ . i { ∈ { 1 j 1 j+1 i} ∈ } Add (i, 0) to X if G[ v , . . . , v ] Γ. Let X be the graph with vertex-set V (X) := S X i { 1 i} ∈ i∈N i and all edges of the form (i, j)(i + 1, j), (i 1, 0)(i, i), or (i, 0)(i + 1, 0). We now show that for − i 2, each vertex (i, j) X has a neighbour in X − . If j = 0 then (i 1, 0) is a neighbour of > ∈ i i 1 − (i, j) in X − . Now suppose that j [i 1]. Then G[ v , . . . , v − , v , . . . , v ] Γ. Since Γ i 1 ∈ − { 1 j 1 j+1 i} ∈ is monotone, G[ v , . . . , v − , v , . . . , v − ] Γ, implying that (i 1, j) is a neighbour of { 1 j 1 j+1 i 1} ∈ − (i, j) in X − . Finally, if j = i, then (i 1, 0) is a neighbour of (i, j) in X − . So every vertex i 1 − i 1 in Xi has a neighbour in Xi−1. By Lemma 2.2, X contains an inﬁnite path (1, x1), (2, x2),... where (i, xi) Xi. By construction, for some i N, we have x1 = = xi−1 = 0 and ∈ ∈ 0 ··· i = xi = xi+1 = xi+2 = . For every ﬁnite subgraph G of G vi, there is an integer n, such 0 ··· − that G is a subgraph of G[ v , . . . , v − , v , . . . , v ], which is in Γ. Since Γ is monotone, { 1 i 1 i+1 n} G0 Γ. Since Γ is extendable, G v Γ. Hence G Γ+ and Γ+ is extendable. ∈ − i ∈ ∈

5 Treewidth–Path Product Structure

The primary result of this section describes a graph, the strong product of a bounded treewidth graph and a path, that contains every planar graph, and satisﬁes several of the key properties mentioned in Section1. The result is extended in various ways for graphs embeddable on any ﬁxed surface, for any proper minor-closed class, and for several non-minor-closed graph classes of interest.

5.1 Layered Partitions

A layering of a graph G is an ordered partition (V0,V1,... ) of V (G) such that for every edge vw E(G), if v V and w V , then i j 1. Note that a layering is equivalent to a ∈ ∈ i ∈ j | − | 6 partition whose quotient is a path. Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [63] defined the layered width of a partition of a graph G to be the minimum ` N such that P ∈ for some layering (V ,V ,... ) of G, each part in has at most ` vertices in each layer V . 0 1 P i Dujmović et al. [63] proved the following lemma in the finite case. The straightforward proof also works for infinite graphs.

Lemma 5.1 ([63]). If a graph G has a partition with layered width `, then G is isomorphic P to a subgraph of (G/ ) P∞ K . Conversely, for every graph H and subgraph G of P `

45 H P∞ K , there is a partition of G with layered width at most ` such that G/ is ` P P isomorphic to a subgraph of H.

For a, c N0 and k, ` N, let Γk,c,`,a be the class of graphs isomorphic to subgraphs of ∈ ∈ (( + K ) P∞ K ) + K . Theorem 4.5 and Lemma 5.1 imply: Tk c ` a Lemma 5.2. A graph G is in Γ if and only if there is a set A V (G) of size at most k,c,`,a ⊆ a, and a partition of G A with layered width ` such that tw ((G A)/ ) k. P − c − P 6

The following notation will be helpful to prove that Γ is extendable. If and are k,c,`,a P1 P2 partitions of a set X, then 1 2 if for all A1 1 and A2 2, either A1 A2 = or P ⊆ P ∈ P ∈ P S ∩ ∅ A1 A2. If 1, 2,... are partitions of a set X and 1 2 ... , then n∈ n is the ⊆ P P S P ⊆ P ⊆ N P partition of X, where A is in n if A n for some n N, and for all n N no strict n∈N P ∈ P ∈ ∈ superset of A is . Pn

Lemma 5.3. Γk,c,`,a is extendable

Proof. Let G be a graph such that every finite subgraph of G is in Γk,c,`,a. Let (v1, v2,... ) be a vertex-ordering of G. Let Gn := G[ v1, . . . , vn ] for n N. Let Xn be the set of all { } ∈ triples (A , , ) such that A is a set of at most a vertices in G , is a layering of n Ln Pn n n Ln G A , and is a partition of G A such that L P ` for all L and P , n − n Pn n − n | ∩ | 6 ∈ Ln ∈ Pn and tw ((G A )/ ) k. Since G is in Γ , Lemma 5.2 implies that X = . And c n − n Pn 6 n k,c,`,a n 6 ∅ ( ) := S Xn is finite since Gn is finite. Let Q be the graph with vertex-set V Q n∈N Xn, where each (A , , ) X is adjacent to (A v , v , v ), which is in X − . By n Ln Pn ∈ n n \{ n} Ln − n Pn − n n 1 Lemma 2.2, there is a path ((A , , ), (A , , ),... ) in Q with (A , , ) X for 1 L1 P1 2 L2 P2 n Ln Pn ∈ n all n N. Then A1 A2 ... and 1 2 ... and 1 2 ... . Let A := nAn and ∈ ⊆ ⊆ L ⊆ L ⊆ P ⊆ P ⊆ ∪ := and := S . Then A is a set of at most a vertices in G, is a layering of L ∪nLn Pn n Pn L G A, and is a partition of G A such that L P 6 ` for each L and P . For − P 0 − | ∩ | 0 ∈ L ∈ P every finite subgraph G of (G A)/ , for some n N, G is a subgraph of (Gn An)/ n, − P ∈ − P implying that tw (G0) tw ((G A )/ ) k. By Lemma 4.20, tw ((G A)/ ) k. c 6 c n − n Pn 6 c − P 6 Hence G Γ by Lemma 5.2. Therefore Γ is extendable. ∈ k,c `,a k,c `,a

Theorem 4.5 and Lemma 5.3 imply:

Lemma 5.4. Let G be a graph such that every ﬁnite subgraph of G is isomorphic to a subgraph of ((H + Kc) P ) + Ka for some graph H with treewidth at most k and for some path P . Then (( + K ) P∞) + K contains G. Tk c a

An analogous proof using Theorem 4.13 instead of Theorem 4.5 and Lemma 4.18 instead of Lemma 4.16 gives:

Lemma 5.5. Let G be a graph such that every ﬁnite subgraph of G is isomorphic to a subgraph of ((H + Kc) P ) + Ka for some graph H with simple treewidth at most k and for some path P . Then (( + K ) P∞) + K contains G. Sk c a

46 5.2 Planar Graphs

The starting point for the proof of Theorem 1.2 is the result of Pilipczuk and Siebertz [136], who showed that every planar graph has a partition into geodesic paths whose contraction gives a graph with treewidth at most 8. This result was reﬁned by Dujmović et al. [63] as follows.

Theorem 5.6 ([63]). Every ﬁnite planar graph G is isomorphic to a subgraph of H P , for some planar graph H with treewidth at most 8 and for some path P .

Theorem 5.6 has been used to solve several open problems regarding queue layouts [63], non-repetitive colourings [61], centred colourings [58], clustered colourings [62], adjacency labellings (equivalently, strongly universal graphs) [22, 60, 72], and vertex rankings [25].

Ueckerdt, Wood, and Yi [164] modiﬁed the proof of Theorem 5.6 to establish the following.

Theorem 5.7 ([164]). Every ﬁnite planar graph G is isomorphic to a subgraph of H P , for some planar graph H with simple treewidth at most 6 and for some path P .

Lemma 5.4 and Theorem 5.7 imply the following theorem introduced in Section1.

Theorem 5.8. P∞ contains every planar graph. S6 Dujmović et al. [63] also proved the following variation on Theorem 5.6.

Theorem 5.9 ([63]). Every ﬁnite planar graph G is isomorphic to a subgraph of H P K3 for some planar graph H with tw(H) 6 3 and for some path P .

In Theorem 5.9, since H is planar and tw(H) 6 3, by the above-mentioned result of Kratochvíl and Vaner [115], we have stw(H) 6 3. This can also be seen directly from the proof of Theorem 5.9 and is implicitly mentioned in [63]. Then Lemma 5.4 and Theorem 5.9 imply:

Theorem 5.10. P∞ K contains every planar graph. S3 3 We now show that the graphs described in Theorems 5.8 and 5.10 satisfy properties (K1)–(K3) from Section1. It follows from results of Hickingbotham and Wood [99] that every ﬁnite subgraph of P∞ has minimum degree at most 19 and every ﬁnite subgraph of P∞ K S6 S3 3 has minimum degree at most 34. Note that χ( P∞) 14 and χ( P∞ K ) 24 S6 6 S3 3 6 since χ(A B) χ(A) χ(B). Lemma 2.11 implies that every n-vertex subgraph of P∞ 6 S6 or P∞ K has treewidth O(√n) and has a balanced separation of order O(√n). S3 3 Lemma 2.17 imply that P∞ and P∞ K (since tw( K ) 11) have linear S6 S3 3 S3 3 6 colouring numbers. Together, this says that P∞ and P∞ K satisfy properties S6 S3 3 (K1)–(K3) from Section1.

5.3 Graphs on Surfaces

Dujmović et al. [63] proved the following results for ﬁnite graphs of bounded Euler genus. A graph X is apex if X v is planar for some vertex v. −

47 Theorem 5.11 ([63]). Every ﬁnite graph G of Euler genus g is isomorphic to a subgraph of:

(a) H P Kmax{2g,1} for some apex graph H of treewidth 9 and some path P , (b) H P Kmax{2g,3} for some apex graph H of treewidth 4 and some path P , (c) (K2g + H) P for some planar graph H of treewidth 8 and some path P .

Since stw(G) stw(G v) + 1 for every graph G and vertex v of G, applying Theorem 5.7 6 − instead of Theorem 5.6, the treewidth bounds in Theorem 5.11 become:

Theorem 5.12. Every ﬁnite graph G of Euler genus g is isomorphic to a subgraph of:

(a) H P Kmax{2g,1} for some apex graph H of simple treewidth 7 and some path P , (b) H P Kmax{2g,3} for some apex graph H of simple treewidth 4 and some path P , (c) (H + K2g) P for some planar graph H of simple treewidth 6 and some path P .

Lemma 5.4 and Theorem 5.12 imply:

Theorem 5.13. For every g N0, each of ( 6 + K1) P∞ Kmax{2g,1}, ( 3 + K1) P∞ ∈ S S K and ( + K ) P∞ contain every graph of Euler genus g. max{2g,3} S6 2g

Dujmović et al. [63] proved the following product structure theorem for ﬁnite apex-minor-free graphs.

Theorem 5.14 ([63]). For every ﬁnite apex graph X, there exists c N such that every ∈ ﬁnite X-minor-free graph G is isomorphic to a subgraph of H P for some graph with H with tw(H) 6 c and for some path P .

Lemma 5.4 and Theorem 5.14 imply:

Theorem 5.15. For every ﬁnite apex graph X there exists c N such that c P∞ contains ∈ T every X-minor-free graph G.

Dujmović et al. [62] showed an analogous result for bounded degree graphs excluding an arbitrary ﬁxed minor.

Theorem 5.16 ([62]). For every ﬁnite graph X there exists c N such that for all ∆ N ∈ ∈ every ﬁnite X-minor-free graph with maximum degree ∆ is isomorphic to a subgraph of H P for some graph H with tw(H) 6 c∆ and for some path P .

Lemma 5.4 and Theorem 5.16 imply:

Theorem 5.17. For every graph X there exists c N such that for all ∆ N, the graph ∈ ∈ P∞ contains every X-minor-free graph with maximum degree at most ∆. Tc∆

5.4 Beyond Minor-Closed Classes

A recent direction pursued by Dujmović, Morin, and Wood [64] studies graph product structure theorems for various non-minor-closed graph classes. Here we extend their results

48 for inﬁnite graphs. First consider graphs that can be drawn on a ﬁxed surface with a bounded number of crossings per edge. A graph is (g, k)-planar if it has a drawing in a surface of Euler genus at most g such that each edge is involved in at most k crossings. Even in the simplest case, there are (0, 1)-planar graphs that contain arbitrarily large complete graph minors [59].

Theorem 5.18 ([64]). Every ﬁnite (g, k)-planar graph is isomorphic to a subgraph of H P , for some graph H of treewidth O(gk6) and for some path P .

Lemma 5.4 and Theorem 5.18 imply:

6 Theorem 5.19. For all g, k N there exists an integer c O(gk ) such that c P∞ ∈ ∈ T contains every (g, k)-planar graph.

Map graphs, which are deﬁned as follows, provide another example of a non-minor-closed classes that has a product structure theorem. Start with a graph G0 embedded in a surface of Euler genus g, with each face labelled a ‘nation’ or a ‘lake’, where each vertex of G0 is incident with at most d nations. Let G be the graph whose vertices are the nations of G0, where two vertices are adjacent in G if the corresponding faces in G0 share a vertex. Then G is called a (g, d)-map graph.A (0, d)-map graph is called a (plane) d-map graph; see [39, 75] for example. The (g, 3)-map graphs are precisely the graphs of Euler genus at most g; see [59]. So (g, d)-map graphs generalise graphs embedded in a surface, and we now assume that d > 4 for the remainder of this section. Theorem 5.20 ([64]). Every ﬁnite (g, d)-map graph is isomorphic to a subgraph of:

• H P KO(gd2), where H is a graph with treewidth at most 14 and P is a path, 2 • H P , where H is a graph with treewidth O(gd ) and P is a path.

Lemma 5.4 and Theorem 5.20 imply:

2 Theorem 5.21. For all g, d N there exists c O(gd ) such that c P∞ contains every ∈ ∈ T (g, d)-map graph.

A string graph is the intersection graph of a set of curves in the plane with no three curves meeting at a single point; see [76, 77, 135] for example. For δ N, if each curve is in at most ∈ δ intersections with other curves, then the corresponding string graph is called a δ-string graph.A (g, δ)-string graph is deﬁned analogously for curves on a surface of Euler genus at most g.

Theorem 5.22 ([64]). Every ﬁnite (g, δ)-string graph is isomorphic to a subgraph of H P , for some graph H of treewidth O(gδ7) and some path P .

Lemma 5.4 and Theorem 5.22 imply:

7 Theorem 5.23. For all g, δ N there exists c O(gδ ) such that c P∞ contains every ∈ ∈ T (g, δ)-string graph.

Finally, we mention the following theorem of Dujmović et al. [64]. The k-th power of a graph G is the graph Gk with V (Gk) := V (G), where vw E(Gk) whenever dist (v, w) [k]. ∈ G ∈

49 Theorem 5.24 ([64]). For every ﬁnite graph X there exists c N such that for all k, ∆ N ∈ ∈ and for every ﬁnite X-minor-free graph G with maximum degree ∆, the k-th power Gk is 4 k isomorphic to a subgraph of H P , for some graph H of treewidth k (c∆) and some path P .

Lemma 5.4 and Theorem 5.24 imply:

Theorem 5.25. For every ﬁnite graph X there exists c N such that for all k, ∆ N, 4 k ∈ k ∈ for some integer t k (c∆) , the graph P∞ contains the k-th power G of every 6 Tt X-minor-free graph G with maximum degree ∆.

5.5 Excluding an Arbitrary Minor

Dujmović et al. [63] proved the following product structure theorem for ﬁnite graphs excluding a ﬁxed minor.

Theorem 5.26 ([63]). For every ﬁnite graph X there exist k, a N such that every ﬁnite ∈ X-minor-free graph G can be obtained by clique-sums of graphs G1,...,Gn such that for i [n], for some graph H with treewidth at most k and some path P , G is isomorphic to a ∈ i i i subgraph of (Hi Pi) + Ka.

Theorem 5.27. For every ﬁnite graph X there is a constant c and there is a graph UX that contains every X-minor-free graph, such that every n-vertex subgraph G of UX has treewidth at most c√n and has a balanced separation of order c√n, and colr(G) cr for every r N. 6 ∈

Proof. Lemmas 5.3 and 4.19 implies that (Γk,c,`,a) is extendable for all k, ` N and D ∈ c, a N0. Theorem 5.26 says that every ﬁnite X-minor-free graph is in (Γk,0,1,a) for ∈ D some k, a depending only on X. Thus every X-minor-free graph is in (Γ ). Let D k,0,1,a U := ( P∞) + K . By Lemma 4.8, U contains every X-minor-free graph. Lemmas 2.11 X bTk a X and 2.12 imply that every n-vertex subgraph of UX has treewidth at most c√n and has a balanced separation of order c√n. Lemmas 2.14 and 2.18 imply that UX has linear colouring numbers. Lemma 2.19 implies that UX has linear expansion.

We finish this section by mentioning one more property of the graph UX in Theorem 5.27. DeVos, Ding, Oporowski, Sanders, Reed, Seymour, and Vertigan [51] proved that for every finite graph X and integer k > 2, there is an integer c such that every finite X-minor-free graph (a) has a vertex k-colouring such that the subgraph induced by any k 1 colour classes − has treewidth at most c, and (b) has an edge k-colouring such that the subgraph induced by any k 1 colour classes has treewidth at most c. DeVos et al. [51] called these ‘low treewidth − colourings’. Dujmović et al. [63] showed that this result is a corollary of Theorem 5.26. Their proof also implies that UX has low treewidth colourings. It is interesting that not only does every X-minor-free graph have low treewidth colourings, but there is a single graph that contains all X-minor-free graphs that itself admits low treewidth colourings.

50 6 Avoiding an Inﬁnite Complete Graph Subdivision

This section focuses on key property (K4), which says that a graph that contains every planar graph should contain no Kℵ0 subdivision. We start the section by giving several characterisations of graphs that contain no Kℵ0 subdivision (Section 6.1). We then construct a graph that contains every planar graph, but contains no Kℵ0 subdivision, amongst other key properties. This shows that Theorem 1.1 is best possible in the sense that it cannot be strengthened to conclude that every graph that contains every planar graph must contain a subdivision of Kℵ0 . Our construction is, in fact, stronger in two respects: it works for Kt-minor-free graphs, and actually gives induced subgraphs. Using the same method, we construct a strongly universal treewidth-k graph, and a graph that contains every locally

ﬁnite graph as an induced subgraph, but contains no Kℵ0 subdivision (Section 6.3). We conclude the section by showing that every graph that contains all planar graphs has a subgraph of inﬁnite edge-connectivity that contains all planar graphs (Section 6.4).

6.1 Characterisations

Robertson et al. [145] characterised graphs containing no subdivision of Kℵ0 in terms of simplicial decompositions; see parts (a) and (b) in Theorem 6.1 below. Part (c) is an equivalent formulation of (b) in terms of tree-decompositions; see Lemma 2.6. Diestel [54] gave a simple proof showing that (a) implies (c), thus establishing the characterisation of Robertson et al. [145] as a corollary. Diestel’s proof uses normal spanning trees; see [26, 101, 137] for more on this theme. Parts (d)–(f) in Theorem 6.1 provide a number of equivalent characterisations related to other parts of this paper. In particular, condition (e) is used in Section 6.2 below. Condition (f) is interesting since it is an inﬁnite analogue of r-shallow minors; see Section 2.7. Theorem 6.1. The following are equivalent for any countable graph G:

(a) G contains no subdivision of Kℵ0 ;

(b) G is a spanning subgraph of a chordal graph that contains no Kℵ0 and has a simplicial decomposition in which each simplicial summand is a finite complete graph; (c) G has a tree-decomposition such that for every infinite set X V (G) there exist ⊆ distinct v, w X in no common bag; ∈ (d) G is a spanning subgraph of a chordal graph with no Kℵ0 subgraph; (e) G is a spanning subgraph of a graph G0 that has a finite chordal partition such that 0 P G / has no Kℵ subgraph; P 0 (f) G contains no model of Kℵ0 with branch sets of finite radius.

As explained above (a), (b) and (c) are equivalent. It is immediate that (b) = (d). ⇒

0 Proof. (d) = (e): Assume that G is a spanning subgraph of a chordal graph G with no Kℵ0 ⇒ 0 0 0 0 subgraph. Let be the (trivial) chordal partition v : v V (G ) of G . So G / ∼= G , 0 P {{ } ∈ } P and thus G / has no Kℵ subgraph. P 0

Proof. (e) = (a): Assume that is a ﬁnite chordal partition of G such that G/ contains ⇒ P P

51 no Kℵ0 subgraph. Suppose that V is an infinite set of vertices in G such that no pair of vertices in V are separated by a finite vertex-cut. For each v V , let A be the part of ∈ v containing v. Since each part of is finite, we may assume (by taking a subset) that P P A = A for all distinct v, w V . Suppose that A A / E(G/ ) for distinct v, w V . Let v 6 w ∈ v w ∈ P ∈ S be a minimal subset of V (G/ ) separating A and A . Since G/ is chordal, S is a clique P v w P in G/ , which is finite by assumption. Hence S X is a finite subset of V (G) separating v P X∈S and w, which is a contradiction. Thus A A E(G/ ) for all distinct v, w V . Therefore v w ∈ P ∈ A : v V induces Kℵ in G/ . This contradiction shows that there is no infinite set { v ∈ } 0 P V of vertices in G such that each pair of vertices in V are separated by no finite cut. In

particular, G contains no Kℵ0 subdivision.

Proof. (f) = (a): If G contains a subdivision of Kℵ , then G contains a model of Kℵ with ⇒ 0 0 branch sets of ﬁnite radius.

Proof. (a) = (f): Assume that G contains a model (Xi)i∈N of Kℵ0 such that Xi has finite ⇒ S radius for each i N. We may assume that V (G) = V (Xi), and that each Xi is ∈ i∈N minimal with the above properties. So each Xi is a tree with finite radius, and every leaf of X is adjacent to some vertex in X for some j = i. Say X is clean if: i j 6 i (i) for each j N i , there is a vertex vi,j Xi adjacent to some vertex in Xj, ∈ \{ } ∈ (ii) there is a vertex ri of infinite degree in Xi, such that for all distinct j, k N with ∈ j, k > i, the rivi,j-path and the rivi,k-path in Xi only intersect at ri.

Suppose that X1,...,Xi−1 are clean for some i N. Since (Xi)i∈ is a model of a complete ∈ N graph, for each j N with j < i, there is a vertex vi,j Xi adjacent to some vertex in Xj. ∈ ∈ Since X has finite radius, some vertex r V (X ) has infinite degree in G. By the minimality i i ∈ i of X , for some infinite set I i + 1, i + 2,... , for distinct j, k I, the r v -path and the i ⊆ { } ∈ i i,j rivi,k-path in Xi only intersect at ri. Replace Xi by the minimal subtree of Xi containing the union of the r v -paths in X , taken over all j 1, . . . , i 1 I. Replace (X ) ∈ by i i,j i ∈ { − } ∪ i i N (Xi)i∈{1,...,i}∪I . In this sequence, X1,...,Xi are clean.

Repeat this step to obtain an inﬁnite Kℵ0 -model (Xi)i∈N in which every Xi is clean. We may assume that vi,jvj,i is an edge of G for all distinct i, j N. We now construct a subdivision ∈ of Kℵ0 in G rooted at (ri2 )i∈ . For i, j N with i < j, let Pi,j be the path from ri2 to N ∈ vi2,j2+i in Xi2 , followed by the edge vi2,j2+ivj2+i,i2 , followed by the path from vj2+i,i2 to vj2+i,j2 in Xj2+i, followed by the edge vj2+i,j2 vj2,j2+i, followed by the path from vj2,j2+i to rj2 in Xj2 . Note that the Pi,j are internally disjoint (since i < j and ` < k and j < k implies 2 2 j + i < k + `). So G contains a Kℵ0 subdivision.

6.2 Kt-Minor-Free Graphs

The main result of this section, Theorem 6.6, says that for every integer t > 3 there is a graph U that contains every Kt-minor-free graph as an induced subgraph, and U satisﬁes key properties (K1), (K3) and (K4). The main tool is that of a chordal partition, which have been used by several authors in the study of Kt-minor-free graphs [12, 140, 148, 165, 166]. Chordal partitions are useful since if is a connected chordal partition of a K -minor-free P t

52 graph G, then the quotient G/ is a minor of G, implying G/ has no K subgraph P P t and tw(G/ ) t 2 by Lemma 4.4. Thus, using chordal partitions, the structure of P 6 − Kt-minor-free graphs can be described in terms of much simpler graphs of treewidth at most t 2. We also use the machinery developed to construct a strongly universal treewidth-k − graph (Proposition 6.3) and a graph that contains every locally ﬁnite graph as an induced

subgraph, but contains no Kℵ0 subdivision (Proposition 6.7).

All these results rely on the following deﬁnitions. For n N0, let n = Hn,1,Hn,2,... be S ∈ H { } a class of ﬁnite graphs. Let := n. Let := (m, n): m N0, n N, m < n . H n∈N0 H X { ∈ ∈ } For (m, n) , say (J, A, B) is an ( , m, n)-triple if: ∈ X H • J is a graph with V (H) = A B and A B = ; ∪ ∩ ∅ • J[A] is isomorphic to some graph in ; and Hm • J[B] is isomorphic to some graph in . Hn If (J ,A ,B ) and (J ,A ,B ) are ( , m, n)-triples, then write (J ,A ,B ) = (J ,A ,B ) 1 1 1 2 2 2 H 1 1 1 ∼ 2 2 2 if there is an isomorphism φ : V (J ) V (J ) such that φ(A ) = A and φ(B ) = B . An 1 → 2 1 2 1 2 -system is a set = (Jm,a,n,b,c,Am,a,Bn,b):(m, n) , a, b, c N such that for all H J { ∈ X ∈ } (m, n) and a, b, c N, ∈ X ∈ • (J ,A ,B ) is an ( , m, n)-triple; m,a,n,b,c m,a n,b H • J[Am,a] ∼= Hm,a and J[Bn,b] ∼= Hn,b; • if c = 1 then there are no edges between Am,a and Bn,b in Jn,a,m,b,c.

For any -system and k N, let (k) be the set of all graphs G with the following H J ∈ J properties:

• there is a partition of G and a tree T with V (T ) = rooted at some part R ; P P ∈ P • there is a (k + 1)-colouring c of T such that G/ is a spanning subgraph of Q := T c P h i (which implies tw(G/ ) tw(Q) k by Lemma 4.4); and P 6 6 • there is a labelling of Q, such that: – for each part A , if dist (R,A) = m and A is labelled a, then G[A] = H ; ∈ P T ∼ m,a and – for every oriented edge AB E(Q), ∈ ∗ if distT (R,A) = m and A is labelled a N and distT (R,B) = n and B is ∈ labelled b N and AB is labelled c N, then ∈ ∈ (G[A B], A, B) = (J ,A ,B ) ∪ ∼ m,a,n,b,c m,a n,b

(which implies that G[A] ∼= Hm,a and G[B] ∼= Hn,b); and ∗ if AB E(Q) E(G/ ) then c = 1. ∈ \ P

Lemma 6.2. For every class = Hm,a : m N0, a N of ﬁnite graphs, for every H { ∈ ∈ } -system = (Jm,a,n,b,c,Am,a,Bn,b):(m, n) , a, b, c N , and for every k N, H J { ∈ X ∈ } ∈ (k) • no graph in contains a subdivision of Kℵ0 ; (k) J • has a strongly universal graph UJ ; J ,k • if, for some b N, every graph in is b-colourable, then every graph in (k) is ∈ H J b(k + 1)-colourable; and • if, for some d, r N, every graph in is d-degenerate, and for all (m, n) and ∈ H ∈ X

53 a, b, c N, in the graph Jm,a,n,b,c, each vertex in Bn,b has at most r neighbours in Am,a, ∈ then every ﬁnite subgraph of any graph in (k) is (rk + d)-degenerate. J

Proof. Let G (k). Let be the partition of G and let Q be the supergraph of G/ ∈ J P P witnessing that G (k). Let G0 be the graph obtained by adding edges to G so that A B ∈ J ∪ is a clique in G0 for each edge AB E(Q). (This makes sense since A, B V (G).) Now, ∈ ⊆ P is a connected partition of G0 with quotient Q, so is a chordal partition. Since every graph P in is finite, is a finite chordal partition. By Theorem 6.1(e), G0 and thus G contains no H P Kℵ0 subdivision. This proves the first claim. Let be the universal tree rooted at an arbitrary vertex r (see Theorem 4.1). Let c be a T (k + 1)-colouring of , such that each vertex v of has infinitely many children of each T T colour distinct from the colour assigned to v. So = c is the universal treewidth-k Tk T h i graph. By Lemma 4.6, there is a k-orientation and labelling of , such that for every chordal Tk graph Z with no Kk+2 subgraph, for every rooted k-orientation of Z, and for every labelling of Z, there is an orientation-preserving label-preserving isomorphism from Z to a subgraph of . Tk The graph U = UJ is obtained from as follows. For each vertex v of at distance m ,k Tk T from r in and labelled a N, replace v by a copy of Hm,a with vertex set Uv V (U), T ∈ ⊆ where U U = for all distinct v, w V ( ). Then, for each oriented edge vw of v ∩ w ∅ ∈ Tk Tk labelled c N, if (m, n) := (distT (r, v), distT (r, w)) and v is labelled a N and w is ∈ ∈ X ∈ labelled b N, then add edges between Uv and Uw so that ∈ (U[U U ],U ,U ) = (J ,A ,B ). (1) v ∪ w v w ∼ m,a,n,b,c m,a n,b So U : v V ( ) is a partition of U, where takes the role of Q in the above definition { v ∈ Tk } Tk of (k). So U (k). J ∈ J We now show that every graph G in (k) is isomorphic to an induced subgraph of U. J Let , T , R, c and Q = T c be as in the definition of (k). Thus, there is a rooted P h i J k-orientation and labelling of Q, such that for every oriented edge AB E(Q), if (m, n) := ∈ (distT (R,A), distT (R,B)) and A is labelled a N and B is labelled b N and AB is ∈ X ∈ ∈ labelled c N, then ∈ (G[A B], A, B) = (J ,A ,B ), (2) ∪ ∼ m,a,n,b,c m,a n,b and if AB E(Q) E(G/ ) then c = 1. ∈ \ P Let Q0 be a chordal graph containing Q as a spanning subgraph, with a k-orientation that preserves the given k-orientation of Q, and subject to these conditions is edge-maximal. Transfer the labels of the vertices and edges of Q to Q0. Label each edge of Q0 E(Q) by 1. − By Lemma 4.7, there is an orientation-preserving label-preserving isomorphism φ from Q0 to an induced subgraph of k. Thus, for every part A labelled a N, we have φ(A) is a T ∈ P ∈ vertex in labelled a. Let Tk " # [ UG := U Uφ(A) . A∈P Since φ is orientation-preserving and label-preserving, for every oriented edge AB E(Q0) ∈ labelled c N, if A is labelled a N and B is labelled b N, then φ(A)φ(B) is an edge in ∈ ∈ ∈

54 labelled c, and Tk (G[A B], A, B) = (J ,A ,B ) = (U[U U ],U ,U ). (3) ∪ ∼ m,a,n,b,c m,a n,a ∼ φ(A) ∪ φ(B) φ(A) φ(B) Note that to conclude that G[A B] = J in the case that AB E(Q0) E(Q), we ∪ ∼ m,a,n,b,c ∈ \ use the fact that AB is labelled c = 1, implying that there is no edge between Am,a and Bn,b in J (by the deﬁnition of -system). m,a,n,b,1 H For each vertex v of G, if A is the part of containing v, then let ψ(v) be the vertex P that v is mapped to in Uφ(A) under the isomorphism described in (3). So ψ(v) is in UG. For every edge vw E(G), if v A and w B , then A = B or AB E(Q), ∈ ∈ ∈ P ∈ ∈ P ∈ implying ψ(v)ψ(w) E(U ) by the isomorphisms in (3). So G is isomorphic to a spanning ∈ G subgraph of UG. We claim that in fact G is isomorphic to UG. Consider an edge xy of UG. So x U and y U for some A, B V (Q). Since φ(Q0) is an induced subgraph of ∈ φ(A) ∈ φ(B) ∈ , either A = B or AB E(Q0). Thus ψ−1(x)ψ−1(y) is an edge of G by the isomorphisms Tk ∈ in (3). Hence G is isomorphic to UG, which proves the second claim. If every graph in is b-colourable, then χ(G/ ) tw(G/ ) + 1 k + 1, and a product H P 6 P 6 colouring gives a b(k + 1)-colouring of G. This proves the third claim.

For the fourth claim, each graph in has an acyclic orientation with in-degree at most d H at each vertex. Say G (k). Let and Q be as in the deﬁnition of (k). So Q has a ∈ J P J k-orientation. Each A has in-degree at most k in Q, and the subgraph G[A] (which is ∈ P isomorphic to a graph in ) has an acyclic orientation with in-degree at most d at each H vertex. For each edge AB E(Q), orient each AB-edge of G from A to B. By assumption, ∈ each vertex in B has at most r neighbours in A. In total, the in-degree of each vertex in G is kr + d. Thus each ﬁnite subgraph of G has minimum degree at most kr + d.

Proposition 6.3. For each k N there is a strongly universal treewidth-k graph. ∈

0 Proof. Let := m := K1 for each m N0. Let J be the graph with vertex set H H { } ∈ v, w and edge-set vw . Let J 00 be the graph with vertex set v, w and edge-set . For { } { } { } 0 ∅ (m, n) and a, b, c N, let Am,a := v and Bn,b := w ; let Jm,a,n,b,c := J if c is even, ∈ X 00 ∈ { } { } and let Ja,b,c := J if c is odd. So = (Jm,a,n,b,c,Am,a,Bn,b):(m, n) , a, b, c N is J { ∈ X ∈ } an -system. For every graph G in (k), if is the partition of G in the deﬁnition of (k), H J P J then since each part is a singleton, tw(G) = tw(G/ ) k. P 6 We now show that every graph G of treewidth k is in (k). By Lemma 4.4, G is a spanning J subgraph of a chordal graph Q = T c with no K subgraph, for some spanning tree h i k+2 T of Q rooted at R, and for some (k + 1)-colouring c of T . Let be the partition of P G and of Q with one part for each vertex. So is a chordal partition of Q. Label P every vertex of G/ by 1. Fix a k-orientation of Q. Consider an oriented edge AB of P Q. Let m := dist (R,A) and n := dist (R,B). If AB is in G/ , then label AB by 2, so T T P (G[A B], A, B) = (J 0, v , w ) = (J ,A ,B ). If AB is not in G/ , then label ∪ ∼ { } { } m,1,n,1,2 m,1 n,1 P AB by 1, so (G[A B], A, B) = (J 00, v , w ) = (J ,A ,B ). Since J 0 has an ∪ ∼ { } { } m,1,n,1,1 m,1 n,1 edge and J 00 does not, is a partition of G, where T and Q satisfy the deﬁnition of (k). P J Thus G is in (k). J Hence (k) is exactly the set of all graphs with treewidth at most k. By Lemma 6.2, (k) J J

55 has a strongly universal graph UJ ,k, which therefore is a strongly universal treewidth-k graph.

The following deﬁnition and lemmas are used in our proof for Kt-minor-free graphs below. If S is a set of k 2 vertices in a connected graph G and r S, then a set X V (G) is an > ∈ ⊆ S-connector (rooted at r) if there are geodesic paths P1,...,Pk−1 in G starting at r, such − that S X = Sk 1 V (P ), and G[X] is a connected subgraph with bandwidth at most k 1, ⊆ i=1 i − and every vertex in G X has at most 2k 2 neighbours in X. An S-connector is called an − − `-connector for any integer ` S . > | | Lemma 6.4 ([166]). For any set S of at least two vertices in a connected graph G, and for any vertex r S, there is an S-connector in G rooted at r. ∈

We now deﬁne a speciﬁc -system to be used in Lemma 6.5. This system is designed so that H Lemma 2.13 can be used to show that graphs in (t−2) have bounded generalised colouring J numbers.

Fix t N with t > 3. As illustrated in Figure6, for m N0, let m be the class of all ∈ m+1 ∈ H connected ﬁnite graphs H with V (H) N0 [t 2], where each v V (H) is denoted by m+1 ⊆ × − ∈ v = (v0, v1, . . . , vm, v?) N [t 2], such that: ∈ 0 × − • for all v, w V (H) if v = w and v = w , then v = w; ∈ m m ? ? • there is exactly one vertex r V (H) with r = 0; ∈ m • for every edge vw E(H) and for every i [0, m], we have v w 1; ∈ ∈ | i − i| 6 • for each i [t 2], the set r v V (H): v = i induces a path in H (which is a ∈ − { } ∪ { ∈ ? } geodesic by the previous rules); and • H has bandwidth at most t 2. − (1,3) (2,3) (3,3) (4,3) (5,3)

(1,2) (2,2) (3,2) (4,2) (5,2) (6,2) (7,2)

r =(0,1) (1,1) (2,1) (3,1) (4,1) (5,1) (6,1)

Figure 6: A graph in with t = 5. Red edges must be present. Gray edges are possibly H0 present.

Note that is countable. Enumerate = H ,H ,... . Let := S . For Hm Hm { m,1 m,2 } H m∈N0 Hm (m, n) , let be the set of all ( , m, n)-triples (J, A, B) such that: ∈ X Jm,n H • J is a graph with V (J) = A B and A B = ; ∪ ∩ ∅ • J[A] and J[B] ; ∈ Hm ∈ Hn • for all v A and w B, if vw E(J) then v w 1 for each i [0, m]; and ∈ ∈ ∈ | i − i| 6 ∈ • every vertex in B has at most 2t 4 neighbours in A. −

56 Note that is countable. Let := S . Enumerate = Jm,n J (m,n)∈X Jm,n J (Jm,a,n,b,c,Am,a,Bn,b):(m, n) , a, b, c N so that it is an -system. { ∈ X ∈ } H The next lemma is inspired by an analogous result for ﬁnite graphs by van den Heuvel and Wood [166].

Lemma 6.5. Every K -minor-free graph G is in (t−2). t J

Proof. Since contains infinitely many disjoint copies of as an induced subgraph, it Tk Tk suffices to assume that G is connected. We may also assume that G is infinite. Let u1, u2,... be an arbitrary enumeration of V (G). Below we define ( i,Ti,Si, ci)i∈ , where for i N, P N ∈ (P1) is a connected chordal partition of G (that is, G/ is a chordal graph, and G[A] is Pi Pi connected for each A ); ∈ Pi (P2) T is a spanning tree of G/ and c is a (t 1)-colouring of T such that G/ is a i Pi i − i Pi spanning subgraph of T c ; ih ii (P3) S is a subtree of T with exactly i nodes, and every node of T V (S ) is a leaf of T ; i i i − i i (P4) S S S ; 1 ⊆ 2 ⊆ · · · ⊆ i (P5) for j [i] and for each part A , we have c (A) = c (A); ∈ ∈ Pj j i (P6) both S and T are rooted at part R where R := u ; i i ∈ Pi { 1} (P7) for each part B V (Si), if m := distSi (R,B) and R = X0,X1,...,Xm = B is the ∈ + RB-path in S , and B is the vertex-set of the component of G (X X − ) i − 0 ∪ · · · ∪ m 1 containing B, then • B is a (t 1)-connector in G[B+] rooted at some vertex r ; and − B • G[B] is isomorphic to some graph in , where each vertex v B is mapped to Hm ∈ (v0, v1, . . . , vm, v?), where vj = dist + (rXj , v) for each j [0, m]; and G[Xj ] ∈ (P8) for each directed edge AB of G/ where A, B V (S ), if (m, n) := Pi ∈ i (dist (R,A), dist (R,B)) , then the triple (G[A B], A, B) is isomorphic to Si Si ∈ X ∪ some ( , m, n)-triple in , H Jm,n (P9) for each directed edge AB of G/ where A V (S ) and B V (T ) V (S ), each Pi ∈ i ∈ i \ i vertex in B has at most 2t 4 neighbours in A. − We first define ( ,T ,S , c ). Let (B ) ∈ be the vertex-sets of the components of G u . P1 1 1 1 j j J − 1 Let be the partition of G with parts R := u and (B ) ∈ . So G/ is a star with J P1 { 1} j j J P1 | | leaves. Let T1 be this star. Colour R by 1 and colour each node Bj by 2. So T1 is a spanning tree of G/ and c is a (t 1)-colouring of T such that G/ = T = T c . Let S be P1 1 − 1 P1 1 1h 1i 1 the tree with one vertex R. Consider T and S to be rooted at R. Then ( ,T ,S , c ) 1 1 P1 1 1 1 satisfies the above conditions.

Note that (P1)–(P3) imply that the nodes of S and T are parts of . So (P4) implies that i i Pi for j [i] each part of that is in S is also a part of in S . Note that (P4) also implies ∈ Pj j Pi i that B+ (deﬁned in (P7)) does not depend on i.

Assume that ( ,T ,S , c ),..., ( ,T ,S , c ) are deﬁned. Since V (S ) = i and each part of P1 1 1 1 Pi i i i | i | in S is ﬁnite, some vertex of G is in a part of that is a node of T V (S ). Let j be the Pi i Pi i − i minimum integer such that u is in a part A that is a node of T V (S ). So A is a leaf j ∈ Pi i − i of T by (P3). Let B ,...,B be the neighbours of A in G/ . So A, B ,...,B is a clique i 1 s Pi { 1 s} in G/ , implying s [0, t 2]. Let B be the parent of A in T . So B ,...,B are ancestors Pi ∈ − s i 1 s

57 of A in Ti. Let m := distTi (R,Bs). Let R = X0,X1,...,Xm = Bs be the RBs-path in Ti. + For ` [0, m], let X be the vertex-set of the component of G (X X − ) containing ∈ ` − 0 ∪ · · · ∪ ` 1 X . By (P7), X is a (t 1)-connector in G[X+] rooted at some vertex r . Note that ` ` − ` X` A X+. ⊆ ` For p [s], let a be any vertex in A adjacent to some vertex in B , which exists since ∈ p p ABp E(G/ i). Let C be a uj, a1, . . . , as -connector in G[A] rooted at uj, which exists ∈ P { } Ss by Lemma 6.4. Let P1,...,Ps be the corresponding geodesic paths. So C = p=1 V (Pp), and C is a (t 1)-connector in G[A]. − Let be the partition of G obtained from by replacing A by C and the vertex sets Pi+1 Pi of the components of G[A] C. For p [s], since a C, we have B C is an edge of − ∈ p ∈ i G/ . So C,B ,...,B is a clique in G/ . For each component X of G[A] C, the Pi+1 { 1 s} Pi+1 − neighbourhood of V (X) in G/ is a subset of C,B ,...,B . So G/ is chordal. By Pi+1 { 1 s} Pi+1 construction, each part of is connected. Thus (P1) is satisﬁed for i + 1. Pi+1 Let Ti+1 be the tree obtained from T by renaming the node A by C, and by adding, for each component G[X] of G[A] C, the node X and the edge XC to T . So X is a leaf in − i+1 Ti+1. Let ci+1 be the colouring of Ti+1 obtained from ci as follows. Let ci+1(C) := ci(A). So C has the same in-neighbourhood in T c as A in T c . For each component G[X] i+1h i+1i ih ii of G[A] C, the neighbourhood of X in G/ is a subset of the clique C,B ,...,B − Pi+1 { 1 s} of size at most t 2 (since G has no K minor). Let c (X) be a colour not used by its − t i+1 neighbours in G/ , which exists since there are t 1 colours. Since B A is an edge of Pi+1 − p G/ for each p [s], if B X is an edge of G/ , then B X is an edge of T c . Thus Pi ∈ p Pi+1 p i+1h i+1i G/ is a spanning subgraph of T c , and (P2) holds for i + 1. (P5) holds by the Pi+1 i+1h i+1i deﬁnition of ci+1.

Let Si+1 be the tree obtained from Si by adding a new node C adjacent to Bs. Thus (P3), (P4) and (P6) are satisﬁed for i + 1.

We now show that (P7) holds for i + 1. By construction, C is a (t 1)-connector in G[A] and − A = C+ (where C+ is the vertex-set of the component of G (X X ) containing C). − 0 ∪ · · · ∪ m Thus C is a (t 1)-connector in G[C+]. For each v C, let v := dist (u , v), and for − ∈ m+1 G[A] j each ` [0, m], let v` := distG[X+](r`, v). Thus, v` w` 6 1 for each edge vw in G[C] and ∈ ` | − | for each ` [m + 1]. For each v C, let v be an integer p [s] such that v is in P . Since ∈ ∈ ? ∈ ? p P is a geodesic, if v = w and v = w , then v = w. Thus v C : v = p induces p m+1 m+1 ? ? { ∈ ? } a path in H, namely P . This shows that G[C] is isomorphic to a graph in , and (P7) p Hm+1 holds for i + 1.

We now show that (P8) holds for i + 1. Consider p S. Let ` := dist (R,B ). Thus G[B ] ∈ Ti p p is isomorphic to some graph in . By (P9), each vertex in C has at most 2t 4 neighbours H` − in Bp. For v Bp and w C and j [`] we have vj = dist + (rB , v) (by (P7)) and ∈ ∈ ∈ G[Bp ] p wj = dist + (rBp , w) (by the deﬁnition of wj); so if vw E(G) then vj wj 6 1. Hence G[Bp ] ∈ | − | (G[B C],B ,C) is isomorphic to some ( , `, m + 1)-triple. This shows that (P8) holds for p ∪ p H i + 1.

Finally, by Lemma 6.4, every vertex in G[A] C has at most 2t 4 neighbours in A, implying − − (P9) is satisﬁed for i + 1.

58 This shows that ( ,T ,S , c ),..., ( ,T ,S , c ) satisfy (P1)–(P9), which com- P1 1 1 1 Pi+1 i+1 i+1 i+1 pletes the deﬁnition of ( ,T ,S , c ), ( ,T ,S , c ),... . P1 1 1 1 P2 2 2 2 By (P1)–(P4), S1,S2,... are trees, where each node of Si is a part of i, and S1 S2 ... . 0 P ⊆ 0 ⊆ For i N, let be the subset of i corresponding to parts in Si. Thus a part of is also ∈ Pi P Pi a part of 0 for all ` i; that is, 0 0. Let := S 0. We claim that is a partition P` > Pi ⊆ P` P i∈N Pi P of G. By construction, each vertex of G is in at most one part of . Consider vertex u of P j G. If u is no part of 0 , then each of the j nodes of S contains a vertex u with ` < j j Pj j ` (by the choice of connector roots), which is a contradiction. So u is in some part of 0 . j Pj := S Hence is a partition of G. Let S i∈N Si. Then S is a spanning tree of G/ rooted P P 0 at R = u1 . For each part A , let c(A) := ci(A) for any i N for which A . The { } ∈ P ∈ ∈ Pi choice of i is irrelevant by (P5). Every edge of G/ is an edge of Si ci for some i N, so P h i ∈ G/ is a spanning subgraph of S c . For each part A , if m := dist (R,A), then by P h i ∈ P S (P7), G[A] = Hm,a for some a N; label A by a. For each oriented edge AB E(G/ ) with ∼ ∈ ∈ P (m, n) := (dist (R,A), dist (R,B)) , by (P8) the triple (G[A B], A, B) is isomorphic S S ∈ X ∪ to some triple (J ,A ,B ) ; label AB by c. Therefore G (t−2) where m,a,n,b,c m,a n,b ∈ Jm,n ∈ J Q := G/ . P Theorem 6.6. For each integer t > 3, there is a graph U such that:

• U contains every Kt-minor-free graph as an induced subgraph,

• U contains no subdivision of Kℵ0 , • U is (t 1)2-colourable, − • U is (t 2)(2t 3)-degenerate, − − • colr(U) (t 2)(t 1)(2r + 1) for every r N, 6 − − ∈

Proof. Let and be as defined prior to Lemma 6.5. By Lemma 6.2, there is a strongly- H J (t−2) universal graph U := UJ − in that contains no subdivision of Kℵ . By Lemma 6.5, ,t 2 J 0 every K -minor-free graph G is in (t−2). So G is isomorphic to an induced subgraph of U. t J Since χ(H) bw(H) + 1 for every graph H, every graph in is (t 1)-colourable. Thus 6 H − the third part of Lemma 6.2 is applicable with b = t 1, implying U is (t 1)2-colourable. − − By assumption, the fourth part of Lemma 6.2 is applicable with d = t 2 (since degeneracy − is at most bandwidth) and k = t 2 and r = 2t 4. Thus every finite subgraph of U is − − (2t 3)(t 2)-degenerate. − − We now show that col (U) (t 2)(t 1)(2r + 1). Let be the (t 2)-oriented chordal r 6 − − P − partition of U from Lemma 6.2. Let H be a finite subgraph of U. Let be the set of Y parts in that intersect V (H). Let H+ be the subgraph of U induced by the union of all P the parts in that intersect V (H). Since H is finite and each part of is finite, H+ is Y Y finite. Note that H+/ is an induced subgraph of U/ . So is a chordal partition of H+, Y P Y and the (t 2)-orientation of U/ defines a (t 2)-orientation of H+/ . Since is finite, − P − Y Y there is an ordering Y ,...,Y of the parts in , such that i < j for every oriented edge 1 n Y Y Y E(H+/ ).(Y ,...,Y is called a ‘perfect elimination’ ordering in the literature on i j ∈ Y 1 n chordal graphs.)

Consider Y . There are at most t 2 neighbouring parts Y N + (Y ) with j < i i ∈ Y − j ∈ H /Y i (and they form a clique in H+/ , although we will not need this property). Let H+ be the Y i

59 + + connected component of H (Y1 Yi−1) that contains Yi. Observe that Hi is precisely + − ∪· · ·∪ the subgraph of H induced by the union of all parts Yj such that i = j or Yj is a + ∈ Y descendent of Yi in H / . By the deﬁnition of , V (Yi) is the union of the vertex-sets of Y + H + t 2 geodesic paths P1,...,Pt−2 in H [Yi]. We claim that P1,...,Pt−2 are geodesics in Hi . − + Let x and y be vertices in some Pj. Let P be any xy-path in Hi . Say P has length `. By the deﬁnition of n and m,n, for each edge vw in P , we have vi wi 6 1. Thus xi yi 6 `, H J | − +| | − | implying distPj (x, y) 6 `. Thus P1,...,Pt−2 are geodesics in Hi . Hence Lemma 2.13 is applicable to H+ with d = p = t 2. Thus col (H) col (H+) (t 2)(t 1)(2r + 1). By − r 6 r 6 − − Lemma 2.16, col is extendable. Therefore col (U) (t 2)(t 1)(2r + 1). r r 6 − −

6.3 Locally Finite Graphs

We have shown above that there exists a graph that contains every planar graph as an induced

subgraph, but contains no subdivision of Kℵ0 . Indeed, the result holds for Kt-minor-free graphs. We now show the same conclusion holds for all locally ﬁnite graphs.

Proposition 6.7. There is a graph that contains every locally ﬁnite graph as an induced

subgraph, but contains no subdivision of Kℵ0 .

Proof. Let be the class of all finite graphs. Let be the class of all ( , m, n)-triples, H J H where (m, n) . So and are countable. Enumerate and so that is an -system. ∈ X (1) H J H J J H By Lemma 6.2, has a strongly universal graph UJ ,1 that contains no subdivision of J 1 Kℵ . We now prove that every locally finite graph G is in . 0 J Let G1,G2,... be the connected components of G. For i N, let ri be any vertex of Gi. ∈ For j N0, let Vi,j be the set of vertices in Gi at distance j from ri. Note that Vi,0 = ri . ∈ { } Fix i N. We prove by induction on j N0 that Vi,j is finite. The base case holds since ∈ ∈ V = 1. Assume that V is finite. Let ∆ be the maximum degree of vertices in V . | i,0| i,j i,j i,j Thus V ∆ V , and V is finite. | i,j+1| 6 i,j | i,j| i,j+1 So = Vi,j : i N, j N0 is a partition of G. The quotient G/ is the disjoint union of P { ∈ ∈ } P infinite 1-way paths (one for each component of G). Let Q be the graph obtained from G/ P by adding an edge between Vi,0 and Vi+1,0 for each i N. So Q is a tree, which has a rooted ∈ 1-orientation obtained by directing the edges from Vi,j to Vi,j+1, and from Vi,0 to Vi+1,0.

For i N and j N0, since Vi,j is ﬁnite, G[Vi,j] = Ha for some a N. Label Vi,j ∈ ∈ ∼ ∈ by a. If Vi,j is labelled a and Vi,j+1 is labelled b, then there exists c N such that ∈ (G[V V ],V ,V ) = (J ,H ,H ) ; label the edge V V of Q by c. For i,j ∪ i,j+1 i,j i,j+1 ∼ a,b,c a b ∈ J i,j i,j+1 i N, if Vi,0 is labelled a and Vi,j+1 is labelled b, then label the edge Vi,0Vi+1,0 of Q by 1. ∈ Thus (G[V V ],V ,V ) = (J ,H ,H ) since there are no edges between i,0 ∪ i+1,0 i,0 i+1,0 ∼ a,b,1 a b ∈ J Aa,b,1 and Ba,b,1 in Ja,b,1. Together, this shows that G (1), and therefore G is isomorphic to an induced subgraph ∈ J of UJ ,1.

60 6.4 Forcing Inﬁnite Edge-Connectivity

As shown above, there is a graph U that contains all planar graphs, but does not contain a subdivision of an infinite clique. In particular, U does not contain a subgraph of infinite vertex-connectivity. We now show that the analogous statement for edge-connectivity is false. That is, infinite edge-connectivity is unavoidable in any graph that contains all planar graphs.

Proposition 6.8. If a graph U contains all planar graphs, then U contains a subgraph of inﬁnite edge-connectivity that contains all planar graphs.

Proof. We first observe that every finite or countably infinite planar graph G is an induced subgraph of a planar graph I(G) of infinite edge-connectivity. To see this, first extend G to a connected planar graph in which G is an induced subgraph. For each edge xy of G, add a new vertex of degree 2 adjacent to x and y. Call the resulting graph G1. For each edge xy of G1, add a new vertex of degree 2 adjacent to x and y. Call the resulting graph G . Continue like this defining G ,G ,... and put I(G) := G G G .... Then I(G) 2 3 4 ∪ 1 ∪ 2 ∪ contains G as an induced subgraph and has infinite edge-connectivity.

Now assume U contains all planar graphs. Let U 0 be the union of all planar subgraphs of U with infinite edge-connectivity. Then U 0 has no finite edge-cut. Hence each component of U 0 has infinite edge-connectivity. By definition, U 0 contains all planar graphs of infinite 0 0 edge-connectivity (and hence all planar graphs by the initial observation). Let U1,U2,... be the components of U 0. It remains to be proved that one of these contains all planar graphs. 0 Suppose for the sake of contradiction that Ui does not contain the planar graph Mi for each 0 i N. Let M be the disjoint union of M1,M2,.... Then I(M) is not a subgraph of any U ∈ i and hence not a subgraph of U 0, a contradiction which proves the theorem.

Proposition 6.8 holds more generally for any minor-closed class of graphs containing the series-parallel graphs (that is, the graphs containing no K4 minor). It also holds for the class of outerplanar graphs if, in the proof, we only add vertices of degree 2 adjacent to the end-vertices of edges on the outer-cycle.

7 Final Remarks

We conclude with several remarks and open problems.

The Primary Question: What is the simplest graph that contains all planar graphs? We have shown that P∞ and P∞ K contain all planar graphs. It is a tantalising open S6 S3 3 problem whether 3 P∞ contains all planar graphs. Note that for each ` N, Dujmović S ∈ et al. [63] constructed a planar graph (with simple treewidth 3) that is not a subgraph of 2 P∞ K`. More generally, for all k, ` N, Dujmović et al. [63] constructed a graph with T ∈ simple treewidth k that is not a subgraph of − P∞ K . Tk 1 `

61 Minimality: Does there exist a graph U0 that contains every planar graph and is a subgraph of every graph that contains every planar graph? Note that U0 must be a common subgraph of Kℵ ℵ ℵ ℵ , P∞, P∞ K , and the graph in Theorem 6.6 (with t = 5); 0, 0, 0, 0 S6 S3 3 it must also contain Kℵ0 as a minor.

Acyclic Colourings: One way to measure the ‘quality’ of a graph that contains every planar graph is via the acyclic chromatic number. The result of Kierstead and Yang [104] mentioned in Section 2.6 along with Lemma 2.17 implies

χa( P ) col ( P ) 5(k + 1). (4) Tk 6 2 Tk 6

With Theorem 5.8 this implies that P∞ contains every planar graph and is acyclically S6 35-colourable. What is the minimum c such that there exists an acyclically c-colourable graph that contains every planar graph? Borodin [24] proved that every ﬁnite planar graph has an acyclic 5-colouring. An easy application of Lemma 2.2 shows that acyclic chromatic number is extendable. Thus every planar graph has an acyclic 5-colouring. More generally, Theorem 5.13 and (4) imply that ( + K ) P∞ contains every graph of Euler genus g S6 2g and has an acyclic (10g + 35)-colouring. Alon, Mohar, and Sanders [7] proved that graphs with Euler genus g are acyclically O(g4/7)-colourable (and this bound is tight up to a polylogarithmic factor). Is there a graph that contains every graph of Euler genus g and has acyclic chromatic number o(g)? The analogous questions for game chromatic number or oriented chromatic number are also of interest.

List Colouring: Given d N, is there a universal graph for the class of d-list-colourable ∈ graphs? For d = 5, any such graph must contain every planar graph.

Bounded Degree Planar Graphs: If a graph U contains all planar graphs with maximum degree 3, must U have inﬁnite maximum degree? Or is there a graph with bounded maximum degree that contains all planar graphs with maximum degree 3? More generally, is there is a graph with bounded maximum degree that contains all graphs with maximum degree 3 (regardless of planarity)? Of course, the above questions are of interest with 3 replaced by any constant.

Minor-Closed Classes: Diestel et al. [56] ﬁrst addressed the question of which minor- closed classes have a universal element. Theorem 1.1(a) implies that every proper minor-closed class that contains every planar graph has no universal element. It is open whether every proper minor-closed class excluding some ﬁnite planar graph has a universal element. The following result is in this direction:

Theorem 7.1. The following are equivalent for a minor-closed class : G (a) has finite treewidth, G (b) some graph with finite treewidth contains every graph in , G (c) some finite planar graph is not in . G

62 Proof. First, (a) implies (b) since for every k N there is a universal graph for the class of ∈ treewidth k graphs (Theorem 4.5).

To see that (b) implies (c), assume that some graph with treewidth n N contains every ∈ graph in . Since treewidth is subgraph-closed, every graph in has treewidth at most n. G G Thus, the (n + 1) (n + 1) grid graph, which has treewidth n + 1, is not in . That is, some × G ﬁnite planar graph is not in , as desired. G Finally, (c) implies (a) by the Grid-Minor Theorem of Robertson and Seymour [144] and since treewidth is extendable (Lemma 4.16).

Acknowledgements

Thanks to Kevin Hendrey, Daniel Mathews, and Alex Scott for helpful comments.

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70 Index

G + uv, 25 k-ended, 21 t Kt , 30 Euler genus, 10 T c , 34 extendable, 10, 11, 43 h i (k), 53 J finite,8, 10 acyclic,9, 17 forbidden pattern, 43 acyclic chromatic number, 17 geodesic,9 adhesion, 16 graph class,9 ancestor,9, 11 graph parameter, 11 apex, 47 k-apex, 31 hereditary,9 apex set, 31 in-degree,9 Apollonian networks, 38 in-neighbourhood,9 bags, 12 infinite, 44 balanced, 15 infinite path,9 bandwidth, 12 isomorphic,9 bounded expansion, 20 f-isoperimetric, 22 bounding function, 20 join,6 branch set, 10 jump, 23

Cartesian product,5 labelling, 33 child, 11 layered width, 45 chord, 10 layering, 45 chordal, 10 -limit, 22 G chromatic number,9 linear colouring numbers, 17 clean, 52 linear expansion, 20 clique,9 locally ﬁnite,9 clique-number,9 locally Hamiltonian, 20 colouring,9 (g, d) k-colouring,9 -map graph, 49 d r-colouring number, 16 -map graph, 49 column, 25 minimal, 10 connected, 10 minor,9 S-connector, 56 minor-closed,9 model,9 `-connector, 56 monotone,9 contains,4 contains an H-subdivision, 10 near-triangulation, 20 countable,8,9 neighbourhood,9 degeneracy,9 order, 15 d-degenerate,9 orientation,9 degree,9 1-orientation, 11 descendant,9, 11 k-orientation, 14 direct product,5 out-degree,9

71 out-neighbourhood,9 simple treewidth, 38 outer cycle, 20 simplicial k-orientation, 14 over, 16 simplicial decomposition, 12 over Γ, 16 simplicial orientation, 13 simplicial summand, 12 parent, 11 stacked polytopes, 38 part, 10 (g, δ)-string, 49 partition, 10 string graph, 49 vw-path,9 δ-string graph, 49 path-decomposition, 12 strong product,5 pathwidth, 12 strongly sublinear separators, 15 planar,3 strongly universal,7 (g, k)-planar, 49 subdivision, 10 planar 3-trees, 38 subgraph,9 planar triangulation, 20 surrounds, 26 plane graph, 20 -system, 53 plane triangulation, 20 H polynomial expansion, 20 thick, 41 power, 49 tight, 26 proper,9 torso, 16 protecting near-triangulation, 24 tree, 11 quotient, 10 tree-decomposition, 12 treewidth, 12 Rado graph,7 triangulation of, 20 random graph,7 ( , m, n)-triple, 53 H k-regular,9 trivial,8 rooted, 11, 14 rooted r-colouring number, 17 universal,4

separating, 10 vertex-partition, 10 separation, 15 k-separation, 15 wheel centred at v, 20 r-shallow minor, 19 width, 12 k-simple, 38 -width, 44 F