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Tree-Decompositions in Finite and Infinite Graphs

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Tree-Decompositions in Finite and Infinite Graphs Tree-decompositions in finite and infinite graphs Dissertation zur Erlangung des Doktorgrades an der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der Universit¨atHamburg vorgelegt im Fachbereich Mathematik von Johannes Carmesin Hamburg 2015 To Sarah Contents Introduction 5 0.1 End-preserving spanning trees . .5 0.2 Canonical tree-decompositions . .6 0.3 Infinite matroids of graphs . .7 0.3.1 Approach 1: Topological cycle matroids . .8 0.3.2 Approach 2: Matroids with all finite minors graphic . .8 0.4 Harmonic functions on infinite graphs . .9 0.5 Acknowledgements and basis of this thesis . 10 1 All graphs have tree-decompositions displaying their topologi- cal ends 11 1.0.1 Introduction . 11 1.0.2 Definitions . 12 1.0.3 Example section . 14 1.0.4 Separations and profiles . 17 1.0.5 Distinguishing the profiles . 25 1.0.6 A tree-decomposition distinguishing the topological ends . 31 2 Canonical tree-decompositions 37 2.1 Connectivity and tree structure in finite graphs . 37 2.1.1 Introduction . 37 2.1.2 Separations . 41 2.1.3 Nested separation systems and tree structure . 45 2.1.4 From structure trees to tree-decompositions . 47 2.1.5 Extracting nested separation systems . 54 2.1.6 Separating the k-blocks of a graph . 57 2.1.7 Outlook . 65 2.2 Canonical tree-decompositions of finite graphs I. Existence and algorithms . 66 2.2.1 Introduction . 66 2.2.2 Separation systems . 67 2.2.3 Tasks and strategies . 73 2.2.4 Iterated strategies and tree-decompositions . 83 2 2.3 Canonical tree-decompositions of finite graphs II. Essential parts . 87 2.3.1 Introduction . 87 2.3.2 Orientations of decomposition trees . 88 2.3.3 Bounding the number of inessential parts . 91 2.3.4 Bounding the size of the parts . 96 2.4 A short proof of the tangle-tree-theorem . 101 2.4.1 Introduction . 101 2.4.2 Preliminaries . 101 2.4.3 Proof . 101 2.5 k-Blocks: a connectivity invariant for graphs . 102 2.5.1 Introduction . 102 2.5.2 Terminology and background . 103 2.5.3 Examples of k-blocks . 104 2.5.4 Minimum degree conditions forcing a k-block . 106 2.5.5 Average degree conditions forcing a k-block . 112 2.5.6 Blocks and tangles . 114 2.5.7 Finding k-blocks in polynomial time . 115 2.5.8 Further examples . 120 2.5.9 Acknowledgements . 121 2.6 Canonical tree-decompositions of a graph that display its k-blocks 122 2.6.1 Introduction . 122 2.6.2 Preliminaries . 123 2.6.3 Construction methods . 127 2.6.4 Proof of the main result . 135 3 Infinite graphic matroids 140 3.1 Infinite trees of matroids . 140 3.1.1 Introduction . 140 3.1.2 Preliminaries . 142 3.1.3 A simpler proof in a special case . 145 3.1.4 Simplifying winning strategies . 148 3.1.5 Presentations . 149 3.1.6 Trees of presentations . 151 3.1.7 (O2) for trees of presentations . 155 3.1.8 (IM) for trees of presentations . 157 3.2 Topological cycle matroids of infinite graphs . 160 3.2.1 Introduction . 160 3.3 Preliminaries . 162 3.3.1 Ends of graphs . 166 3.3.2 Proof of Theorem 3.2.4 . 169 3.3.3 Consequences of Theorem 3.2.4 . 174 3.4 Matroids with all finite minors graphic . 177 3.4.1 Introduction . 177 3.4.2 Preliminaries . 179 3.4.3 Graph-like spaces . 181 3 3.4.4 Pseudoarcs and Pseudocircles . 184 3.4.5 Graph-like spaces inducing matroids . 188 3.4.6 Existence . 190 3.4.7 A forbidden substructure . 197 3.4.8 Countability of circuits in the 3-connected case . 199 3.4.9 Planar graph-like spaces . 203 4 Every planar graph with the Liouville property is amenable 205 4.0.10 Introduction . 205 4.0.11 Preliminaries . 207 4.0.12 Known facts . 209 4.0.13 Roundabout-transience . 211 4.0.14 Square tilings and the two crossing flows . 213 4.0.15 Harmonic functions on plane graphs . 219 4.0.16 Proof of the main result . 221 4.0.17 Applications . 221 4.0.18 Further remarks . 224 A 234 A.1 Summary . 234 A.2 Zusammenfassung . 234 A.3 My contributions . 235 4 Introduction In Chapters 1 and 2, we build tree-decompositions that display the global struc- ture of infinite and finite graphs. These tree-decompositions of infinite graphs are an important tool to study infinite graphic matroids, which are the topic of Chapter 3. Chapter 4 is independent of the others and contains results on harmonic functions on infinite graphs. 0.1 End-preserving spanning trees In 1931, Freudenthal introduced a notion of ends for second countable Hausdorff spaces [63], and in particular for locally finite graphs1 [64]. These ends are intended as `points at infinity' that compactify the graph when it is locally finite (ie, locally compact). The compacification is similar to the familiar 1- point compactification of locally compact Hausdorff spaces but finer: the two- way infinite ladder, for example, has two such points at infinity, one at either `end', see Figure 1. Figure 1: The two-way infinite ladder has two ends indicated at as the two thick points on the very left and the very right side. Independently, in 1964, Halin [70] introduced a notion of ends for graphs, taking his cue directly from Carath´eodory's Primenden of simply connected regions of the complex plane [33]. For locally finite graphs these two notions of ends agree. For graphs that are not locally finite, Freudenthal's topological definition still makes sense, and gave rise to the notion of topological ends of arbitrary graphs [54]. In general, this no longer agrees with Halin's notion of ends, although it does for trees. Halin [70] conjectured that the end structure of every connected graph can be displayed by the ends of a suitable spanning tree of that graph. He proved 1A locally finite graph is one in which all vertices have finite degree 5 this for countable graphs. Halin's conjecture was finally disproved in the 1990s by Seymour and Thomas [95], and independently by Thomassen [102]. In Chapter 1, we shall prove Halin's conjecture in amended form, based on the topological notion of ends rather than Halin's own graph-theoretical notion. We shall obtain it as a corollary of the following theorem, which proves a conjecture of Diestel [49] of 1992 (again, in amended form): Theorem 1. Every graph has a tree-decomposition (T; ) of finite adhesion such that the ends of T define precisely the topological endsV of G. See Section 3.3 for definitions. We use Theorem 1 as a tool to show that the topological cycles of any graph together with its topological ends induce a matroid, see Section 0.3 below. The tree-decompositions constructed for the proof of Theorem 1 are based on earlier versions for finite graphs, which are a central technique in the following section. 0.2 Canonical tree-decompositions One approach for understanding the global structure of mathematical objects such as graphs or groups is to decompose them into parts which cannot be further decomposed, and to analyse how those parts are arranged to make up the whole. Here we shall decompose a k-connected graph into the `(k+1)-connected pieces'; and the global structure will be tree-like. The idea is modelled on the well-known block-cutvertex tree, which for k = 1 displays the global structure of a connected graph `up to 2-connectedness'. Extending this to k = 2, Tutte proved that every finite connected graph G has a tree-decomposition of adhesion 2 into `3-connected minors' [105]. Chapter 2 is about extending this result to higher connectivities. One way to define k-indecomposable objects is the following: a (k + 1)- block in a graph is a maximal set of at least k + 1 vertices, no two of which can be separated in the ambient graph by removing at most k vertices. We prove that every finite graph has a (canonical) tree-decomposition of adhesion at most k such that any two different (k + 1)-blocks are contained in different parts of the decomposition [42]. Under weak but necessary conditions, these tree-decompositions can be combined into a single tree-decomposition that dis- tinguishes all the (k + 1)-blocks for all k simultaneously. We call (k + 1)-blocks satisfying this necessary condition robust, see Section 2.1 for details. Another notion of highly connected pieces in a graph is that of tangles. These were introduced by Robertson and Seymour in [94] and are a central notion in their theory of graph minors. With the same proof as that of the aforementioned theorem, one can construct a tree-decompositions that does not only distinguish all the (robust) blocks but also all the tangles. This implies and strengthens an important result of the Graph Minors Project of Robertson and Seymour [94]. An important feature of our tree-decompositions is that they are invariant under the group of automorphisms of the graph, whereas theirs is not. Our 6 techniques also allow us to give another simpler proof of the original result of Robertson and Seymour, see Section 2.4. Hundertmark [78] introduced k-profiles, which are a common generalisation of k-blocks and tangles of order k. Together with Lemanczyk [79], he used the proof of the decomposition theorem of [42] in order to construct a tree- decomposition that distinguishes all (robust) profiles.
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