Case Analysis of a Splitter Theorem

Masaryk University Faculty of Informatics

Case analysis of a splitter theorem

Bachelor’s Thesis

Matúš Hlaváčik

Brno, Spring 2016 Replace this page with a copy of the official signed thesis assignment and the copy of the Statement of an Author. Declaration

Hereby I declare that this paper is my original authorial work, which I have worked out on my own. All sources, references, and literature used or excerpted during elaboration of this work are properly cited and listed in complete reference to the due source.

Matúš Hlaváčik

Advisor: prof. RNDr. Petr Hliněný, Ph.D.

i Acknowledgement

I would like to express my deepest gratitude to my advisor, prof. RNDr. Petr Hliněný, Ph.D., for his patience, helpful comments and suggestions. I would also like to thank my family and friends for their support provided not only while working on this thesis, but during all my life.

ii Abstract

After many years of research C. Chun, D. Mayhew, and J. G. Oxley published a new splitter theorem which we could use in an algorithm for finding internally 4-connected graphs which can have planar emulators. The purpose is to limit the possible counter-examples to Fellow’s planar emulator conjecture. In this work, we explain everything needed for understanding this new splitter theorem (contains pieces of graph theory, matroids theory and specific structures of matroids in graphs). This splitter theorem is originally for internally 4-connected binary matroids and is used to generate minors, so in this thesis we describe the version of the theorem for graphs and we turn the theorem "inside out". This enables us to generate bigger graphs while preserving some minor-related properties which are needed for generating possible counter-examples.

iii Keywords graph, matroid, finite planar cover, finite planar emulator, internally 4-connected, two disjoint k-graphs, splitter theorem

iv Contents

1 Introduction 1

2 Basic definitions 2 2.1 Graphs ...... 2 2.2 Matroids ...... 7

3 Introduction of the main problem 9 3.1 Planar covers ...... 9 3.2 Planar emulators ...... 10 3.3 Current state of the problem ...... 11

4 Importance of a splitter theorem 17

5 The new splitter theorem 20 5.1 Operations from Theorem 5.1 ...... 21 5.2 The duals of the operations ...... 22

6 Application of the splitter theorem 39 6.1 Non-useful steps (iv),(v) ...... 41 6.2 Algorithm ...... 41

7 Conclusion 44

Bibliography 45

v List of Figures

2.1 Example of a graph. 2 2.2 Examples of basic classes of graphs (Kn, Pn, Cn, Km,n). 3 2.3 The minor minimal obstruction for the projective plane. [8, 9] 6 2.4 G and G* (dashed) [6]. 8 3.1 (a) Graph G. (b) Planar cover of G. (c) Planar emulator of G. 10 3.2 The graph G = K5 (left) and his planar cover (right) [10]. 12 3.3 An example of a graph having two disjoint k-graphs [4]. 13 3.4 The planar emulator (right) of K4,5 − 4K2 (left) by Rieck and Yamashita [17]. 14 3.5 The graphs G and F (top) and 1-, 2- and 3-expansion of G by F (bottom line). 15 4.1 The operations of splitting a vertex, triad addition and triangle explosion [4]. 18 4.2 Quadrangular, pentagonal and hexagonal extension [4]. 19 5.1 (a) An augmented 4-wheel. (b) The result of deleting the central-cocircuit of an augmented 4-wheel. 24 5.2 (a) A ladder structure (n = 2). (b) The result of a ladder-compression move. 24 5.3 (a) An open rotor chain. (b) The result of trimming an open rotor chain. 25 5.4 (a) A ladder structure. (b) A result of trimming a ladder structure. 25 5.5 (a) A ladder structure. (b) A result of trimming a ladder structure. 26 5.6 (a) An open quartic ladder. (b) The result of a mixed ladder move. 27 5.7 (a) An enhanced quartic ladder. (b) The result of an enhanced ladder move. 28

vi 5.8 (a) An enhanced quartic ladder. (b) The result of an enhanced ladder move. 29 5.9 (a) A bowtie ring. (b) The result of trimming a bowtie ring. 29 5.10 (a) The dual of an augmented 4-wheel. (b) The result of dual of deleting the central-cocircuit of an augmented 4-wheel. 30 5.11 (a) The dual of a ladder structure (n = 2). (b) The result of the dual of a ladder-compression move. 30 5.12 (a) The dual of an open rotor chain. (b) The result of the dual of trimming an open rotor chain. 30 5.13 (a) The dual of a ladder structure. (b) A result of the dual of trimming a ladder structure. 31 5.14 (a) The dual of a ladder structure. (b) A result of the dual of trimming a ladder structure. 32 5.15 (a) The dual of a ladder structure. (b) A result of the dual of trimming a ladder structure. 33 5.16 (a) The dual of an open quartic ladder. (b) The result of the dual of a mixed ladder move. 34 5.17 (a) The dual of an enhance quartic ladder. (b) A result of the dual of an enhanced ladder move. 35 5.18 (a) The dual of an enhance quartic ladder. (b) A result of the dual of an enhanced ladder move. 36 5.19 (a) The dual of an enhance quartic ladder. (b) A result of the dual of an enhanced ladder move. 37 5.20 (a) The dual of a bowtie ring. (b) A result of the dual of trimming a bowtie ring. 38 5.21 (a) The dual of a bowtie ring. (b) A result of the dual of trimming a bowtie ring. 38

6.1 (a) Planar embedding of graph K2,2,2. (b) Planar embedding of graph Q3 (the cube). 40

vii 1 Introduction

A graph is a collection of nodes (vertices) and its connections (edges). Its representation is usually a drawing. We interpret nodes as points and a connection of two nodes as lines between two connected nodes (points). Our often asked question is if we are able to draw a graph without an intersecting of some lines. In other words, if a graph can be embedded in a plane. We are easily able to solve this question because of Kuratowski’s theorem (2.2) [1]. However, what will happen if we allow multiplicity of vertices, but with the condition that we have to preserve the local structure of the graph? This is the question which arises from two independent studies in the 1980s by S. Negami (planar cover) and M. Fellows (planar emulator). Their studies are different for example in the definition of "the local structure", but surprisingly, their main conjectures are same. By the time, there were published some partial results which moved Negami’s conjecture closer to the proof. Nevertheless, Fellows’s conjecture had been stuck for 20 years until the end of 2008, when there was published a new surprising result and Fellows’s conjecture was disproved. In this thesis, we are following the approach used by P. Hliněný in [2] and then by M. Derka in [3] and [4]. We are explaining theory needed to understand a new splitter theorem and also restate the theorem for our purpose. We want to be able to generate all possible counter-examples to Fellow’s planar emulator conjecture. After this introduction follows chapter with basic definitions and theory which is needed to understand before you can read the rest of this work. Chapter 3 is an introduction of the main problem and its terms and also the historical development of the main problem. In Chapter 4 we are explaining what is a splitter theorem and why it is so important for our research. Chapter 5 is presenting the new splitter theorem with theory needed to fully understand and in Chapter 6 there is a sketch of an algorithm for generating all possible counter- examples to Theorem 3.6.

1 2 Basic definitions

This chapter is about basic theory of graphs and matroids. More information about graph theory can be found in [2] and about matroids in [5–7].

2.1 Graphs

A simple undirected graph is a pair G = (V, E), where V is a vertex set and E is an edge set. Edge e is a pair of vertices {u, v}. The vertex set of graph G referred as V(G) and the edge set as E(G). An edge e between u and v (e joins u and v) is denoted by {u, v} or also only uv. When vertices are joined by an edge then they are adjacent or we call them neighbours. Also, we can say the vertices are ends of edge e and they are incident with e. Graphs are often represented as points (vertices) connected by curves (edges), for example Figure 2.1.

x w

u v e

Figure 2.1: Example of a graph.

We denote degree of a vertex v in a graph G by degG(v) – it equals to the number of edges in G incident with v. Now we present basic classes of graphs (examples in Figure 2.2). We say that a graph is k-regular if all its vertices have degree k. A vertex of degree 3 is called cubic vertex and 3-regular graph is cubic graph.

2 2. Basic definitions

A graph for which every pair of vertices are connected by an edge is called complete graph (clique). A complete graph on n vertices has n (k) edges, is denoted by Kn, and it is a n-1-regular graph. We define path of length n as a set of n + 1 vertices {v1, v2,... , vn, vn+1} joined consecutively by n edges {v1v2, v2v3,..., vnvn+1} and tagged as Pn. Vertices on the start and on the finish have degree 1 and another vertices have degree 2. The cycle of length n ≥ 3 is a set of vertices {v1, v2,..., vn} joined in a cyclic fashion by n edges {v1v2, v2v3,..., vnv1} and represented by Cn. All vertices in a cycle have degree two, so it is a 2-regular graph. We say that graph G is bipartite if vertices can be split into two disjoint sets V1, V2 (partitions), such that V1 ∪ V2 = V(G) and the ends of every edge in E(G) are in different partitions. The complete bipartite graph, called Km,n is bipartite graph with set of edges E(Km,n) = {v1v2 | v1 ∈ V1, v2 ∈ V2} (|E(Km,n)| = m · n) and |V1| = m, |V2| = n. This can be easy generalized for p-partite graphs

and Kv1,v2,...,vn .

v2 vn+1 vn

v3 v1

v3 v2 v4 vn v1

v4 v3

v v 5 2 v1 v2 v3 vn

v6 v1

v7 vn u1 u2 um

Figure 2.2: Examples of basic classes of graphs (Kn, Pn, Cn, Km,n).

3 2. Basic definitions

Two graphs G, H are isomorphic, written G ≃ H, if there exists a pair of bijections φ : V(G) → V(H), ψ : E(G) → E(H) such that for every edge uv in E(G) holds φ(u)φ(v) = ψ(uv). Graph H is a subgraph of G if and only if V(H) ⊆ V(G) and E(H) ⊆ E(G), written H ⊆ G. A subgraph H is induced denoted by G H, when all of edges from E(G) with both ends in V(H) is also in E(H). A spanning subgraph is a subgraph H ⊆ G such that V(H) = V(G). Between graph operations which are needed in our thesis belong following. If G is a graph and V′ is a subset of its vertices, then G − V′ (or G ∖ V′) denotes the induced subgraph of G on the vertex set V(G) ∖ V′. In other words the graph G′(V(G) ∖ V′, E(G) ∖ E′) where E′ is the set of all edges which have at least one of ends in V′. If G is a graph and E′ ⊆ E(G) is a set of edges, G − E′ (or G ∖ E′) denotes the graph obtained by deleting all edges of E′ from G. If e is an edge of G then G − e (or G ∖ e) is used as a shortcut for G − {e}. The inverse operation to edge deletion is edge addition. If u, v are two vertices of a graph G, then G + uv denotes the graph obtained by adding a new edge e with ends u, v to G. The graph obtained from G by adding a set of edges E is denoted by G + E. The notation G + e is used as a shortcut for G + {e}. A graph H obtained from a graph G by subdividing an edge e = uv with vertices v1,..., vk that are not in V(G) is H = (G − e) ∪ P, where P and G are disjoint except for the ends u, v. A graph H is a subdivision of a graph G if H is obtained from G by subdividing some edges. Let v ∈ G be a cubic vertex with distinct neighbours. Replacing v by a triangle with vertices in the neighbours of v is called Y∆ transfor- mation. We obtain H from G by contracting an edge e = uv (written F = G/e) if V(H) = V(G) ∖ {u, v} ∪ {w}, E(H) = E(G) ∖ {e}, where w ∈/ V(G). Informally, the edge e = uv is deleted from the graph, and the vertices u, v are identified into vertex w. Following terms are also important for our thesis. Connected graph is a graph G such that for each two vertices u, v from V(G) exist a path in G which ends u and v. A maximal induced and connected subgraph of G is called component of graph G.

4 2. Basic definitions

A (vertex) cut in graph G is a set V ⊂ V(G) such that G − V has more components than G. Cut of size k is a k-cut. A graph G is k- connected if have at least k + 1 vertices, is connected, and minimal cut in G is k−cut. A separation in a graph G is a pair of sets (A, B) such that A ∪ B = V(G) and there is no edge in G between the sets A ∖ B and B ∖ A. A separation (A, B) is nontrivial if both A ∖ B and B ∖ A are nonempty. The order of a separation (A, B) equals |A ∩ B|, and a k-separation is a separation of order k. Clearly, a connected graph has a nontrivial k-separation if and only if it has a k-cut. A graph H is a minor of a graph G if H is obtained from a subgraph of G by a sequence of edge contractions. A graph G is said to have an H-minor if there exists a minor H′ of G such that H′ is isomorphic to H. It can be easily checked that the minor relation is transitive, i.e. if H is a minor of G, and G is a minor of F, then H is a minor of F, too.

Theorem 2.1 (N. Robertson, P. D. Seymour). Let φ be a graph property preserved under taking minors (i.e. if G has φ, then also does every minor of G). There exists finite set of graphs Γ (called forbidden minors) such that G has φ if and only if it does not contain any minor isomorphic to any γ from Γ.

Unfortunately, Theorem 2.1 is non-constructive and therefore, it is not clear how to build set Γ. Exactly for these problems is created splitter theorem. Graphs can be drawn in several different ways. A graph G is said to embed in a surface if it is isomorphic to any drawing GS on S such that all vertices of GD are represented by different points and no edge represented by curve intersect other edges (curve). Such drawing is called embedding. Graphs which are embedded in a plane are called planar and in a projective plane are called projective.

Theorem 2.2 (Kuratowski, 1930 [1]). A graph G is planar if and only if it does not contain a subdivision of K5 or K3,3 as a subgraph. Let G be a graph embedded in a plane. A face of G is each cycle, isomorphic to some Cn, which is the subgraph of G such that all vertices of V(G) ∖ V(Cn) are either inside or outside of this cycle. Size

5 2. Basic definitions

Figure 2.3: The minor minimal obstruction for the projective plane. [8, 9]

of a face fn isomorphic to a cycle Cn is n. The set of all faces of graph G is denoted by F(G). For more information about topology we refer to the source of all previous information [2].

Lemma 2.3 (Euler’s formula). Let G be a planar graph. Then |V(G)| − |E(G)| + |F(G)| = 2.

6 2. Basic definitions

Similarly to planar graphs, we can characterize the set of projective graphs, too (like Theorem 2.2). Glover, Huneke and Wang [8] found a family of forbidden graphs for the projective plane. It is a set Λ of 35 graphs such that each member of Λ is non-projective and is minor minimal with this property (Figure 2.3). Archdeacon proved that the list is complete and those are the only such graphs [9]. Theorem 2.4 (Archdeacon, 1981 [9]). A graph G is projective if and only if it does not contain a subdivision of some graph from family Λ (Figure 2.3) as a minor.

2.2 Matroids

Definition 2.5. A matroid M is an ordered pair (E, ℐ) consisting of a finite set E and collection ℐ of subsets of E satisfying following three conditions: (I1) ∅ ∈ ℐ. (I2) If I ∈ ℐ and I′ ⊂ I, then I′ ∈ ℐ. (I3) If I1, I2 ∈ ℐ and |I1| < |I2|, then there is an element e of I2 ∖ I1 such that I1 ∪ {e} ∈ ℐ. If M is a matroid (E, ℐ), then M is called a matroid on E. The members of ℐ are independent sets of M, and E is the ground set of M. A subset of E that is not in ℐ is called dependent. We should be able to see that a list of the maximal independent sets in a matroid M is clearly more efficient way to specify M than the list of all independent sets. A maximal independent set in M is called bases or a base of M, and the collection of bases is denoted by ℬ or ℬ(M). Lemma 2.6. Let ℬ be a set of subsets of a set E. Then ℬ is a collection of bases of a matroid on E if and only if ℬ satisfies the following conditions: (B1) ℬ is non-empty. (B2) If B1 and B2 are members of ℬ and x ∈ B1 ∖ B2, then there is an element y of B2 ∖ B1 such that (B1 ∖ {x}) ∪ y ∈ ℬ. A minimal dependent set in a matroid M will be called a cicruit of M and we shall denote the set of circuits of M by 풞 or 풞(M). A circuit of M having n elements will also be called an n-cicruit. It is easy to see that 풞(M) can be determined from ℐ(M). Similarly, ℐ(M) can

7 2. Basic definitions

be determined from 풞(M): the members of ℐ(M) are those subsets of E(M) that contain no member of 풞(M). So a matroid is uniquely determined by its set 풞 of circuits.

Lemma 2.7. Let 풞 be a set of subsets of a set E. Then 풞 is the collection of circuits of a matroid on E if and only if 풞 satisfies the following conditions: (C1) ∅ ∈/ 풞. (C2) If C1 and C2 are members of 풞 and C1 ⊆ C2, then C1 = C2. (C3) If C1 and C2 are distinct members of 풞 and e ∈ C1 ∩ C2, then there is a member C3 of 풞 such that C3 ⊆ (C1 ∪ C2) ∖ {e}. Let E be a set of edges of G and 풞 is the set of edge sets of cycles of G. Then the pair (E, 풞) is a matroid. It is called a cycle matroid (also known as graphic matroid) of G and is denoted by M(G). One of the operations on matroids that is important for our work is a making a dual of the matroid. To each matroid M on a set E, there is a dual matroid M* defined on the same set E. The simplest definition of M* is through bases: the bases of M* consists of the complements of the bases of M. A dual of a cycle matroid M on edges of E is the cocycle matroid M*, which is likewise defined on the edges of E. The circuits in M* are the bonds of M and are called the cocircuits of M. For example, the dual of planar graph G is such graph G* that there exists a pair of bijection φ : F(G) → V(G*), ψ : E(G) → E(G*) and for all pairs of vertices u, v in G* hold that they are connected with an edge e if and only if edge ψ−1(e) is on a border between faces φ−1(u) and φ−1(v). For better understanding we can see example in Figure 2.4.

Figure 2.4: G and G* (dashed) [6].

For a closer treatment of matroid theory, we recommend my sources [5–7] or some other publications written by J. G. Oxley.

8 3 Introduction of the main problem

Now we move our attention to the main problem of this thesis, planar emulating. First, we introduce theory about planar covers and emulators and then we describe the state of the problem on the field.

3.1 Planar covers

We start with the definition of planar covers [10]. Definition 3.1. A planar graph H is a planar cover of a graph G if there exists a pair of onto mappings (φ, ψ), φ : V(H) → V(G), ψ : E(H) → E(G), called a (cover) projection, such that ψ maps the edges incident with each vertex v in H bijectively onto the edges incident with φ(v) in G. In other words, every vertex φ(v) has the same degree as v and his neighbours are the images of φ of neighbours of v. From this all we can see, that for every edge e = uv in H holds that the edge ψ(e) in G has endvertices φ(u), φ(v). Hence for simple graphs, it is enough to specify the vertex projection φ that maps the neighbours of each vertex v in H bijectively onto neighbours of φ(v) in G (a traditional approach). The natural question which incoming now is when does a planar cover exist? When we do not have any restrictions then the answer is easy. Imagine some graph G, take one vertex v and start creating a tree. The root v has its neighbours in G as children and each of the children has its neighbours as children except his parent in this tree. We can continue this process in the same way. But this tree is infinite. From now we will consider only finite planar covers. Moreover, since covers for non-connected graphs consist of the covers for their particular components, we can deal with connected graphs only. The interest in graphs with having finite planar cover increased in the 1980s due to Negami, who stated the following: Conjecture 3.2 (Negami, 1988 [11]). A connected graph has a finite planar cover if and only if it embeds in the projective plane. Proposition 3.3 ([10]). If a graph G has a planar cover, then so does every minor of G.

9 3. Introduction of the main problem

Proposition 3.3 allowed us to use Theorem 2.1 as approach to problem of planar covering.

v3 v2

v1 v1

v3 v2 v3 v2 v1

v2 v3 v2 v3

v3 v2 v1 v1

(a) (b) (c)

Figure 3.1: (a) Graph G. (b) Planar cover of G. (c) Planar emulator of G.

3.2 Planar emulators

This section is inspired by [4]. Definition 3.4. A graph H is a planar emulator of a graph G if there exists a pair of onto mappings (φ, ψ), φ : V(H) → V(G), ψ : E(H) → E(G), called an emulator projection, such that ψ maps the edges incident with each vertex v in H (surjectively) onto the edges incident with φ(v) in G. In the case of simple graphs, it is enough to specify the vertex projection φ. Informally speaking, the difference between covers and emulators is that the neighbourhood of a vertex in an emulator may contain “repeated edges”, unlike in a cover where the neighbourhoods must be one-to-one. In other words, the difference between planar covers and emulators is in relaxing the condition that ψ must be bijection to only surjection. That means that having a planar emulator should be "weaker" as- sumption as having a planar cover. An infinite planar emulator of G

10 3. Introduction of the main problem is as easy to build as an infinite planar cove. Therefore, we target our attention to finite planar emulators and also to only planar emulators of connected graphs. This concept was studied by Fellows in 1980s independently of Negami. It is very surprising but Fellows obtained similar result as Negami and gave the following conjecture: Conjecture 3.5 (Fellows, 1988 [12]). There is a finite planar emulator for a graph G if and only if there is a finite planar cover for G. This conjecture was later reformulated by Kitakubo into Conjecture 3.6. Conjecture 3.6 (Kitakubo, 1992 [13]). There is a finite planar emulator for graph G if and only if it embeds in the projective plane. Analogous lemma to Proposition 3.3 holds also for planar emulators: Proposition 3.7 (Kitakubo, 1992 [13]). If a graph G has a planar emulator, then so does every minor of G.

3.3 Current state of the problem

Conjecture 3.2 is still open, but both of Conjectures 3.5, 3.6 were already disproved (see Theorem 3.16). All these following information was collected in [4].

3.3.1 Planar covers In 1980s the problem of planar covers was investigated by S. Negami and D. Archdeacon and later by R.B. Richter, R. Thomas and P.Hliněný. In [14] has been shown that every projective graph has a planar cover and construction of these covers is very simple – the crosscap in a projective-planar drawing of a graph G is replaced with a mirror image of the drawing, and the corresponding connections between the drawings are added (see Figure 3.2). Because every cover is also an emulator, the next claim immediately follows: Proposition 3.8. If graph G has a projective embedding, it also has a finite planar emulator.

11 3. Introduction of the main problem

c2 d2

b a2 c d 2

c1 d1

v1 v

b a

a1 b1

Figure 3.2: The graph G = K5 (left) and his planar cover (right) [10].

To prove Conjectures 3.2 and 3.6 we have to show also the con- verse, i.e. that non-projective graphs have no planar cover and emulator, respectively. Using Propositions 3.3 and 3.7 it is easy to propose following:

Proposition 3.9. The properties of having finite planar emulators, and analogously also planar covers, are closed under taking minors.

Therefore, to prove the conjectures, it is enough to show that there are no finite planar covers and emulators for the minor minimal ob- structions for the projective plane which are known (Figure 2.3).

Proposition 3.10 ([4]). The properties of existence of finite planar covers and emulators are enclosed under taking Y∆ transformations.

The graphs K7 − C4, 풟3 and ℰ5 can be obtained by a sequence of Y∆ transformations from the graph ℱ1. Similarly, the graphs K1,2,2,2, ℬ7, 풞3, 풞4 and 풟2 can be obtained via Y∆ transformations from the graph ℰ2. Thus, the sets ∆Y(K7 − C4) = {K7 − C4, 풟3, ℰ5, ℱ1} and ∆Y(K1,2,2,2) =

12 3. Introduction of the main problem

{K1,2,2,2, ℬ7, 풞3, 풞4, 풟2, ℰ2} are called the families of K7 − C4 and K1,2,2,2 respectively. Some proofs of non-existing planar covers for some of forbidden minors for the projective plane were given: Theorem 3.11 (Archdeacon, Fellows, Hliněný, and Negami, 1988-98). Conjecture 3.2 holds true if and only if there is no finite planar cover for the graph K1,2,2,2 [4].

3.3.2 Planar emulators The concept of finite planar-emulating was proposed by M. Fellows in 1985 [15] and this problem is very tightly connected with Negami’s problem of finite planar-covering [11]. Definition 3.12. Graph G contains two disjoint k-graphs if there exist two vertex-disjoint subgraphs J1, J2 ⊆ G such that, for i = 1, 2, the graph Ji is isomorphic to a subdivision of K4 or K2,3, the subgraph G − V(Ji) is connected and adjacent to Ji, and contracting all the vertices of V(G) ∖ V(Ji) in G into one results in a non-planar graph (i.e. contracting a K5- or K3,3- subdivision). We can see the example of a graph containing two disjoint k-graphs.

Figure 3.3: An example of a graph having two disjoint k-graphs [4].

Theorem 3.13 (Fellows, 1988 [12]). A planar-emulable graph G cannot contain two disjoint k-graphs. Consequently, none of the 19 graphs – connected projective forbidden minors – inbetween the graphs K3,3 · K3,3 and 풢1 (incl.) of Figure 2.3, has a finite planar emulator.

13 3. Introduction of the main problem

Theorem 3.14 (Fellows, 1988 [12]). The graph K3,5 has no finite planar emulator.

Note that because of Euler’s formula, graphs K7 and K4,4 cannot be planar-emulable either. All of these implicate following:

Corollary 3.15. None of the graphs in the family Λ = {K3,3 · K3,3, K5 · K3,3, K5 · K5, ℬ3, 풞2, 풞7, 풟1, 풟4, 풟9, 풟12, 풟17, ℰ6, ℰ11, ℰ19, ℰ20, ℰ27, ℱ4, ℱ6, 풢1, K3,5} (see Figure 2.3) has a finite planar emulator. The graphs K4,4 and K7 are not planar-emulable either. After all of these proofs on non-existence of planar emulators for many of minors forbidden for projective plane, the next result was a big breakthrough, because it falsified Conjectures 3.5 and 3.6.

Theorem 3.16 (Rieck; Yamashita, 2010 [16]). The graphs K1,2,2,2 and K4,5 − 4K2 do have finite planar emulators.

Figure 3.4: The planar emulator (right) of K4,5 − 4K2 (left) by Rieck and Yamashita [17].

Theorem 3.16 is very important because of the planar cover for K4,5 − 4K2 (Figure 3.4) which is not planar-coverable. Therefore, we

14 3. Introduction of the main problem

can claim that the set of planar-emulable graphs is larger as the set of planar-coverable graphs. Note that now K4,5 − 4K2 is not only one graph that has a planar emulator and does not have a planar cover. Finite emulators have been found for ℰ2, 풞4 and all the members of ∆Y(K7 − C4) [17, 18]. From Corollary 3.15 and Theorem 3.16 implicate following:

Corollary 3.17. If a non-projective graph has a finite planar emulator, it must contain a minor isomorphic to one of K4,4 − e, K4,5 − 4K2 or to a graph from the K7 − C4 or K1,2,2,2 families. Definition 3.18 ([2]). A graph G is internally 4-connected if it is simple, 3- connected, has at least five vertices, and for every separation (A, B) of order 3, either G A or G B has at most three edges.

v1 u1

v5 v2 u4 u2

v4 v3 u3

v1 u2

v5 v2 v1 u1 u3

v5 u1 u4 v4 u1

v4 u4 u2 v5 v2 u4 u2

u3 u3 v4 v3

Figure 3.5: The graphs G and F (top) and 1-, 2- and 3-expansion of G by F (bottom line).

15 3. Introduction of the main problem

An n-expansion (n ≤ 3) of a graph G is an embedding of a planar graph F into graph G by identifying of some vertices from G with some vertices of F (example is in Figure 3.5). A graph H is a planar expansion of a graph G if there is a sequence of graphs G0 = G, G1,..., Gi = H such that Gi is a 1-, 2- or 3-expansion of Gi−1 for all i = 1, . . . , i. For formal definition we recommend [4].

Theorem 3.19 (Derka, 2013 [4]). If H is a non-projective graph with finite planar emulator, then H is a planar expansion of an internally 4-connected graph G from a finite set of 176 graphs, or it contains a minor isomorphic to K4,5 − 4K2, ℰ2 or to a graph from the family of K7 − C4.

16 4 Importance of a splitter theorem

A splitter theorem is useful in solving problems of structure of graph theory. It is a technique to generate graphs containing some minor and fulfilling some constraints for which connectivity is important. In other words, it is used to generate possible counterexamples to structure claims, or to show that no counterexample exists. The aim of a good splitter theorem is to define elementary steps which are implementable on computers, exhaustively. A theorem should guarantee that after a small number of steps we have again a graph with the desired connectivity, property, and we are able to check our requested constraints. The most famous splitter theorem is known as Seymour’s Splitter theorem:

Theorem 4.1 (Seymour, 1980 [19]). Let M be a 3-connected matroid with |E(M)| ≥ 4 and let N be a 3-connected proper minor of M. If M is not a wheel or a whirl, then there exists e ∈ E(M) such that M ∖ e or M/e is 3-connected and has an N-minor.

For higher levels of connectivity it is in general very hard to create a useful splitter theorem and this problem is still unsolved. There are many different generalizations of Theorem 4.1 but many of them have some small bugs – for example long sequences of steps without an opportunity for checking constraints (the problem of Theorem 4.2) or impossibility of applying some rules (in specific conditions). The second one can lead to looping (like in Theorem 4.3). For the purpose of finding all possible non-projective graphs with a planar emulator in recent studies [3, 4] Theorems 4.2 and 4.3 (also generalizations of Theorem 4.1) were used. Theorem 4.2 is simplified version of a result proved in [20].

Theorem 4.2 (Johnson; Thomas, 1997). Let G be an internally 4-connected minor of an internally 4-connected graph H such that G has no embedding in the projective plane. Then there exists a sequence J0 = G, J1,... Jk ≃ H of almost internally 4-connected graphs such that for i = 1, 2, . . . k, the graph Ji is obtained from Ji−1 by adding an edge, splitting a vertex, or by a triad addition or by a triangle explosion (Figure 4.1). Moreover, each Ji has at most

17 4. Importance of a splitter theorem

one violating edge and if an edge e is contained in both Ji1 and Ji, it is not violating in at least one of them.

A graph G is called almost internally 4-connected if it is simple, 3- connected, has at least five vertices, and for every separation (A, B) of order 3, either G A or G B has at most four edges. The notion of almost internal 4-connectivity clearly differs from internal 4-connectivity by one edge only. Hence, a pair (v, e), where v is a cubic vertex in G and edge e has both the endvertices adjacent to v, is called a violating pair in G. The edge e in violating pair (v, e) is referred to as a violating edge.

Figure 4.1: The operations of splitting a vertex, triad addition and triangle explosion [4].

Theorem 4.3 (Johnson; Thomas, 2002 [21]). Suppose that G and H are internally 4-connected graphs, G is a proper minor of H, and that G has no embedding in the projective plane. Make a condition that each component of the subgraphs of G induced by cubic vertices is a tree or cycle. Then, either H is an addition extension of G, or there exists a minor H0 of H satisfying one of the following: H0 is a 1-step addition extension of G, or H0 is a quadrangular,

18 4. Importance of a splitter theorem pentagonal or hexagonal extension of G (Figure 4.2), or H0 is obtained by splitting a vertex. The last case happens only if condition is not fullfilled.

Figure 4.2: Quadrangular, pentagonal and hexagonal extension [4].

For correct definitions of the terms from Theorems 4.2 and 4.3we recommend [4].

19 5 The new splitter theorem

Now is published after many years of research a brand new splitter theorem. The theorem brings new opportunities for the results. An unexplained theory in Theorem 5.1 is explained in the rest of this work.

Theorem 5.1 (Chun; Mayhew; Oxley, 2016 [22]). Let G and H be internally 4-connected graphs such that G has a proper H-minor and |E(H)| ≥ 6. Then G has a proper minor G′ such that G′ is internally 4-connected with an H-minor and one of the following holds: (i) |E(G)| − |E(G′)| ≤ 3; or (ii) |E(G) − E(G′)| = 4 and G′ is obtained from G by deleting the central-cocircuit of an augmented 4-wheel, by a ladder-compression move, or by the dual of one of these moves; or (iii) G′ is obtained from G by removing at least four edges by (a) trimming an open rotor chain, a graphic ladder structure, or a ring of bowties, or by the dual of trimming an open rotor chain, a ladder structure, or a ring of bowties; or (b) a mixed ladder move or the dual of such a move; or (c) a graphic enhanced-ladder move or the dual of an enhanced-ladder move; or (iv) G is a quartic Möbius ladder and H is a cubic Möbius ladder with |V(G)| − 1 vertices; or (v) G is Q3 or K2,2,2, and H is K4. Theorem 5.1 is originally for internally 4-connected matroids. But authors of this splitter theorem have also created the version for graphs. The conversion from matroids to graphs may look trivial but it requires not only to specify what the various moves look like for graphs but also what the duals of these moves look like. Our task is to recreate Theorem 5.1 for the purpose of generating all possible internally 4-connected counterexamples for Conjectures 3.2 and 3.6. We know that all of these counterexamples have to be non-projective graphs that do not contain two disjoint k-graphs. We want to start generating of these graphs in the set of forbidden minors for the projective plane for which it is still not obvious if they have a finite planar emulator and continue to bigger graphs.

20 5. The new splitter theorem

However, we can notice, that Theorem 5.1 is formulated in the other direction. They start from a bigger graph and generate its minor which accomplishes constraints. Before reversing Theorem 5.1 we have to fully understand it. There- fore, we explain all terminology needed to understand individual moves and moves on duals. The most of the theory and figures in this chapter are inspired by [22]. Each of circlet vertices in diagrams in this and next chapter corre- sponds to a vertex bond in the graph and no circled vertex equals any other vertex in the graph. While we do not insist that all of the uncircled vertices are distinct, the requirement that the graph G be internally 4-connected imposes constraints. In particular, most of the diagrams have at most four uncircled vertices. If two of these are equal, then there are at most three vertices to which the rest of G is attached, so G can have at most one vertex that is not part of the structure in the diagram. In this chapter we describe individual operations from Theorem 5.1 and its effect on G. We do it in this order (we have listed trimming a ring of bowties in the end of the list because the dual of these operations allows the most variations) and then all its duals: (a) deleting the central cocircuit of an augmented 4-wheel; (b) a ladder-compression move; (c) trimming an open rotor chain; (d) trimming a ladder structure; (e) a mixed ladder move; (f) an enhanced-ladder move; and (g) trimming a ring of bowties.

5.1 Operations from Theorem 5.1

An augmented 4-wheel is represented by the modified graph diagram in Figure 5.1 (a), where the four dashed edges form the central cocircuit. All of the edges in Figure 5.1 are distinct. So deleting the central-cocircuit of an augmented 4-wheel is deleting all dashed edges and the result of this operation can be seen in Figure 5.1 (b). We can see a ladder structure in Figure 5.4 (a). When a graph G contains the structure in Figure 5.2 (a) (it is a ladder structure where

21 5. The new splitter theorem

n = 2) then we say that G ∖ c1, c2/d1, b2 has been obtained from G by a ladder-compression move. We call the structure in Figure 5.3 (a) an open rotor chain. All edges of the structure in the figure are distinct and, for some n ≥ 3, there are n dashed edges. Note that n can be even or odd (no only even as the figure suggests). Operation trimming an open rotor chain is deleting dashed elements from open rotor chain (Figure 5.3). In Figures 5.4 (a) and 5.5 (a) we can see the structures which we call a ladder structures. All edges in these figures are distinct, with excep- tion dn may be the same as γ in 5.5. Either {dn−2, an−1, cn−1, dn−1} or {dn−2, an−1, cn−1, an, cn} is a bond in Figures 5.4 (a) and 5.5 (a). Either {b0, c0, a1, b1} or {β, b0, c0, a1, b1} is also a bond in 5.5 (a). We refer to deleting all dashed edges from Figures 5.4 (a) and 5.5 (a) as trimming a ladder structure. An open quartic ladder is in Figure 5.6. All edges in this figure are distinct except that (a, b, c) may be (d, e, f ) or a may be f . When we delete all dashed edges and contract the edge with arrow we call it a mixed ladder move. We call the structures in Figures 5.7 and 5.8 an enhanced quartic ladder. All edges in these figures are distinct, and we say that deleting all dashed edges is a graphic enhanced-ladder move. Last structure which we bring is a bowtie ring in Figure 5.9 (a). All of the edges in the figure are distinct and the ring contains at least tree triangles. Deleting the dashed edges in Figure 5.9 (a) is called trimming a bowtie ring.

5.2 The duals of the operations

When G contains the structure in Figure 5.10 (a) where all the vertices shown are distinct and {1, 2, 3, 4} is a bond, the graph G = 5, 6, 7, 8 is obtained from G by the dual of deleting the central cocircuit of an augmented 4-wheel. When G contains the structure in Figure 5.11 (a) where all the edges shown are distinct. Then G ∖ c1, c2/d1, b2 is obtained from G by the dual of a ladder-compression move. The dual of trimming an open rotor chain is shown in Figure 5.12. This operation consists of contracting all of the edges in the figure that

22 5. The new splitter theorem

are marked with an arrow. The number of such edges is at least three and may be even or odd. All of the edges in the figure are distinct. We see that this operation turns a cubic ladder segment into a quartic ladder segment. The dual of trimming a ladder structure has several variants de- pending on whether the edges α, β and γ are present in the ladder structure in Figures 5.4, 5.5. These variants are shown in Figures 5.13, 5.14 and 5.15 where the dotted edges, which correspond to α, β and γ, are either all present or all absent. In each of these figures, the move consists of contracting all of the edges marked with an arrow, the number of such edges being at least three. All the edges in each part are distinct. The dual of a mixed ladder move is shown in Figure 5.16. All of the edges are distinct except that a may equal f , or {a, b, c} may equal {d, e, f } in such a way that {b, c} ̸= {d, e}. The move contracts all of the edges marked with an arrow and deletes the dashed edge. There are two variants of an enhanced-ladder move, which arise from the structures shown in Figures 5.7, 5.8 and one variant of an enhanced-ladder move which was not mentioned in previous section. It was not there because the result of the enhanced-ladder move on this ladder do not arise in graphic matroid. However, the duals of all three variants of enhanced-ladder moves do arise in graphic matroids. In Figures 5.17, 5.18, and 5.19 we can see the tree possible structures. In each, all of the edges are distinct and the move contracts all of the edges marked with arrows. The number of such edges being at least three in Figure 5.17, at least four in Figure 5.18, and at least six in Figure 5.19. The dual of a bowtie ring in a graph G consists of a sequence of 4- cycles, {b0, c0, a1, b1}, {b1, c1, a2, b},..., {bn, cn, a0, b0} along with n + 1 distinct cubic vertices u0, u1,..., un where each ui meets ai, bi, and ci. It means that each of the distinguished 4-cycles meets exactly two of the vertices in {u0, u1,..., un} and that these two cubic vertices are non-adjacent in G. We can see two possible variants of the dual of trimming a bowtie ring in Figures 5.20 and 5.21.

23 5. The new splitter theorem

(a) (b)

Figure 5.1: (a) An augmented 4-wheel. (b) The result of deleting the central-cocircuit of an augmented 4-wheel.

d2 d2 d0 d1 d0

a0 c0 a1 c1 a2 c2 a0 c0 a1

b0 b1 b2 b0 b2

(a) (b)

Figure 5.2: (a) A ladder structure (n = 2). (b) The result of a ladder- compression move.

24 5. The new splitter theorem

(a) (b)

Figure 5.3: (a) An open rotor chain. (b) The result of trimming an open rotor chain.

(a)

(b)

Figure 5.4: (a) A ladder structure. (b) A result of trimming a ladder structure.

25 5. The new splitter theorem

(a)

(b)

Figure 5.5: (a) A ladder structure. (b) A result of trimming a ladder structure.

26 5. The new splitter theorem

(a)

(b)

Figure 5.6: (a) An open quartic ladder. (b) The result of a mixed ladder move.

27 5. The new splitter theorem

c2 t0 tk−2 tk−1 tk

a2 b2 d1 c0 u0 v0 vk−2 uk−1 vk−1 uk vk c1 b1

a1 b0 w0 wk−3 wk−2 wk−1

(a)

t0 tk−2 tk−1 tk

a2 b2 d1 u0 uk−2 uk−1 uk b1

a1 b0 wk−3 wk−2 wk−1

(b)

Figure 5.7: (a) An enhanced quartic ladder. (b) The result of an enhanced ladder move.

28 5. The new splitter theorem

c2 t0 tk−3 tk−2 uk−1

a2 tk−1 b2 vk−1 d1 c0 u0 v0 uk−3 vk−3 uk−2 vk−2 wk−1 c1 b1 uk tk a1 s0 sk−3 sk−2 vk

(a)

t0 tk−3 tk−2 uk−1

a2 tk−1 b2 d1 u0 uk−3 uk−2 wk−1 tk b1 uk

a1 s0 sk−3 sk−2

(b)

Figure 5.8: (a) An enhanced quartic ladder. (b) The result of an enhanced ladder move.

(a) (b)

Figure 5.9: (a) A bowtie ring. (b) The result of trimming a bowtie ring.

29 5. The new splitter theorem

(a) (b)

Figure 5.10: (a) The dual of an augmented 4-wheel. (b) The result of dual of deleting the central-cocircuit of an augmented 4-wheel.

(a) (b)

Figure 5.11: (a) The dual of a ladder structure (n = 2). (b) The result of the dual of a ladder-compression move.

(a) (b)

Figure 5.12: (a) The dual of an open rotor chain. (b) The result of the dual of trimming an open rotor chain.

30 5. The new splitter theorem

(a)

(b)

γ b0 b1 bn

a0 a1 an−1 an

α d0 d1 dn

Figure 5.13: (a) The dual of a ladder structure. (b) A result of the dual of trimming a ladder structure.

31 5. The new splitter theorem

(a)

(b)

γ b0 b1 bn−1

a0 a1 an−1 bn

α d0 d1 an dn

Figure 5.14: (a) The dual of a ladder structure. (b) A result of the dual of trimming a ladder structure.

32 5. The new splitter theorem

(a)

(b)

γ a0 b1 bn−1

a1 an−1 α bn

d0 d1 an dn

Figure 5.15: (a) The dual of a ladder structure. (b) A result of the dual of trimming a ladder structure.

33 5. The new splitter theorem

(a)

(b)

α dn d0 a f c a0 a1 an d

b0 b1 bn−1 b e

Figure 5.16: (a) The dual of an open quartic ladder. (b) The result of the dual of a mixed ladder move.

34 5. The new splitter theorem

(a)

(b) d2

d 1 a2 b2 t0 tk

u0 u1 uk

a1 b0 w0 w1 wk

Figure 5.17: (a) The dual of an enhance quartic ladder. (b) A result of the dual of an enhanced ladder move.

35 5. The new splitter theorem

(a)

(b) d2

d 1 a2 b2 t0 tk−1

u0 u1 tk

wk−1

a1 b0 w0 w1 uk wk

Figure 5.18: (a) The dual of an enhance quartic ladder. (b) A result of the dual of an enhanced ladder move.

36 5. The new splitter theorem

(a)

(b) d2

uk−1 d 1 a2 b2 t0 tk−2

u0 u1 tk−1

a w w u 1 b0 0 1 tk k wk−1

Figure 5.19: (a) The dual of an enhance quartic ladder. (b) A result of the dual of an enhanced ladder move.

37 5. The new splitter theorem

a0 a2 a0 a2 1 2 1 2

b0 b1 b2 bn b0 b0 b1 bn 2 1 2 1 a1 c1 a3 a1

(a) (b)

Figure 5.20: (a) The dual of a bowtie ring. (b) A result of the dual of trimming a bowtie ring.

b0 bn b0 bn

b1 bn−1 a0 cn a0 b1 bn−1 3 3 3 3

c a a1 an 0 cn−1 n a 1 c1

(a) (b)

Figure 5.21: (a) The dual of a bowtie ring. (b) A result of the dual of trimming a bowtie ring.

38 6 Application of the splitter theorem

Now we are finally able to restate Theorem 5.1 for our purpose. As we mentioned in the previous chapter, we want to create a version of splitter theorem which allows us to take a small graph H and generate a slightly bigger one with H as a minor. We can repeat this process to generate even bigger graphs.

Observation 6.1. We can turn Theorem 5.1 to a theorem which enables us to generate bigger graph G from G′ with preserving internal 4-connectivity and property of having H-minor.

Let G and H be internally 4-connected graphs such that G has a proper H-minor and |E(H)| ≥ 6. According to the Theorem 5.1 we know that G has a proper G0- minor which we can obtain from G by one of the moves in Theorem 5.1 and also graph G0 is internally 4-connected with an H-minor. Note that |E(G)| > |E(G0)|. If |E(G0)| > |E(H)| then we apply Theorem 5.1 again but now on graph G0 and we get a graph G1 which is internally 4-connected with an H-minor and is the proper minor of G0. Note again that |E(G0)| > |E(G1)|. It is easy to see that with every application of Theorem 5.1 on graph Gi we obtain a graph Gi+1 which is smaller than Gi (because it is his proper minor) and also we can again apply Theorem 5.1 on Gi+1. Because of the fact that the difference between |E(G)| and |E(H)| is a finite number. It is clear that after at most |E(H)| − |E(G)| iteration of applying of Theorem 5.1 we have to get a graph Gk which has |E(H)| vertices. Seeing that |E(Gk)| = |E(H)| and H is a minor of Gk, it must holds that Gk ≃ H. It implicates that H can be obtained from Gk−1 by one of the moves described in Theorem 5.1. It means that graph Gk−1 (which is internally 4-connected with an H-minor) can be obtained from H by the reverse of moves from Theorem 5.1. Because of this, we can use Theorem 5.1 to generate all internally 4- connected graphs G with an H-minor from an internally 4-connected graph G′ with an H-minor by the reverse of the steps from Theorem 5.1.

39 6. Application of the splitter theorem

Theorem 6.2 (reversed Theorem 5.1). Let G and H be internally 4-connected graphs such that G has a proper H-minor and |E(H)| ≥ 6. Then G has a proper minor G′ such that G′ is internally 4-connected with an H- minor and one of the following holds: (i) |E(G′)| − |E(H)| ≤ 3; (ii) |E(G′) − E(H)| = 4 and G′ is obtained from H by adding the central- cocircuit of an augmented 4-wheel, by a ladder-decompression move, or by the dual of one of these moves; (iii) G′ is obtained from H by adding at least four edges by (a) extending an open rotor chain, a graphic ladder structure, or a ring of bowties, or by the dual of extending an open rotor chain, a ladder structure, or a ring of bowties; (b) a reversed mixed ladder move or the dual of such a move; (c) a reversed graphic enhanced-ladder move or the dual of a reversed enhanced-ladder move; (iv) G′ is a quartic Möbius ladder and H is a cubic Möbius ladder with |V(G′)| − 1 vertices; ′ (v) G is Q3 or K2,2,2, and H is K4.

w1 v1

v2 w2

(a) (b)

Figure 6.1: (a) Planar embedding of graph K2,2,2. (b) Planar embedding of graph Q3 (the cube).

40 6. Application of the splitter theorem 6.1 Non-useful steps (iv),(v)

First, we remove from Theorem 6.2 steps which we can not use for our generating internally 4-connected graphs, that extend internally 4-connected non-projective graph, because they extend a planar or projective graphs. A quadrtic Möbius ladder is by definition projective so the item (iv) is for us useless. ′ In item (v) we expect that G (graph which we extend) is Q3 or K2,2,2. Both of them are plannar (see Figure 6.1).

6.2 Algorithm

Now we describe a sketch of an algorithm based on Theorem 6.2 for generating all internally 4-connected graphs without two disjoint k- graphs, that extend given internally 4-connected non-projective graph. We present this algorithm in this section and it is based on three disjoint parts which are based on three options from Theorem 6.2 ((i), (ii), (iii)). At the beginning, we generate all internally 4-connected graphs without two disjoint k-graphs, that extend given internally 4- connected non-projective graph such that (i) from Theorem 6.2 holds, then we generate all internally 4-connected graphs such that (ii) from Theorem 6.2 holds and in the end, and at the end we generate graphs such that (iii) from Theorem 6.2 holds. It is not certain that these moves create two disjoint k-graphs, so we have to test all our generated graphs or containing a two disjoint k-graphs. For the description of this test we reffer to [4].

6.2.1 First option (i) In this option we will generate graphs using Theorem 4.2 (more details about the algorithm are in [3]), but with little change and this change is that we will consider only graphs H with at most 3 edges more than given internally 4-connected non-projective G. This means that we generate all graphs H by sequences J0 = G, J1,... Jk such that following holds: 1. for k = 3 and for i = 1, 2, 3 we obtained Ji from Ji−1 by adding an edge or by splitting a vertex; or

41 6. Application of the splitter theorem

2. for k < 3 we obtained Ji from Ji−1 by ether a triad addition or by a triangle explosion for at most one of i ∈ 1, 2. And then throw all of the graphs which are almost internally 4- connected and on rest of the graphs we run the test for containing a two disjoint k-graphs. The outputs of this option are all of these internally 4-connected graphs H which do not include a two disjoint k-graphs.

6.2.2 Second option (ii) In this part of algorithm, we have to explain individual reversed operation. When a graph contains the structure in Figure 5.1 (b) we say that adding the dashed edges in Figure 5.1 (a) and their central vertex is called adding the central-cocircuit of an augmented 4-wheel. The dual of adding the central-cocircuit of an augmented 4-wheel is the reversed operation to the dual of deleting the central-cocircuit of an augmented 4-wheel (Figure 5.10). We call the reversed operation of a ladder-compression move (Fi- gure 5.2) a ladder-decompression move. Dual of the operation a ladder-decompression move is the reversed of the dual operation of a ladder-compression move (Figure 5.11). When G contains structure S of one of the results of the operations: ∙ deleting the central-cocircuit of an augmented 4-wheel, ∙ a ladder-compression move, or ∙ the dual of one of these moves, then we obtain G′ by the reversed operation, of the operation that can create S, on S. For example when G contains a structure of the result of the dual of a ladder-compression move, than we obtain G′ from G by the dual of a ladder-decompression move on this structure. After we obtain all possible graphs G′ from a given G, we have to run the test for containing a two disjoint k-graphs. The outputs of this part of algorithm are all graphs G which do not contain a two disjoint k-graphs.

42 6. Application of the splitter theorem

6.2.3 Third option (iii) At first, we again have to explain individual reversed operation. The reversed operation of ∙ trimming an open rotor chain (Figure 5.3) is extending an open rotor chain, ∙ trimming a graphic ladder structure (Figures 5.4 and 5.5) is extending a graphic ladder structure, ∙ trimming a ring of bowties (Figure 5.9) is extending a ring of bowties, ∙ a mixed ladder move (Figure 5.6) is reversed mixed ladder move, ∙ a graphic enhanced-ladder move (Figures 5.7 and 5.8) is reversed graphic enhanced-ladder move. Analogously the duals of these operations. Similarly to previous option, when G contains one of the structures S of the results of the operations from Theorem 5.1 (iii), then we obtain G′ by the reversed operation, to the operation that can create S, on S. After we obtain all possible graphs G′ from given G, we have to run the test for containing a two disjoint k-graphs. The outputs of this part of algorithm are all graphs G which do not contain a two disjoint k-graphs.

The output of whole algorithm is the set Γ′ of all graphs generated by all previously mentioned parts. For finding all possible internally 4-connected graphs without two disjoint k-graphs, we have to run the same algorithm on each graph of Γ′ to generate next generation of internally 4-connected graphs without two disjoint k-graphs and repeat it until we do not find them all (the output of iteration of algorithm is empty).

43 7 Conclusion

The aim of all the researches connected with our work is to find all possible counter-examples to Fellow’s planar emulator conjecture. With our approach, we are able to find the set of all possible internally 4-connected counter-examples. The goal of the thesis was to analyse special graph cases of a preprint of Chun, Mayhew and Oxley about a new splitter theorem for internally 4-connected matroids. We were supposed to restrict our attention only to the cases in which the base minor in the theorem is a non-projective graph and the target graph does not contain two disjoint k-graphs. The expected outcome was a description of an algorithm (based on this splitter theorem) for generating all internally 4-connected graphs without two disjoint k-graphs, that extend a given internally 4-connected non-projective graph. All of these goals are accomplished. Besides of it, we also bring together most of the information about this topic and explain the majority of the planar emulator theory. The strong point of this work is that it explains the topic of ad- vanced graph theory not only for experienced readers but also for common students. On the other hand, the weak point of this work is a concreteness of the algorithm, as it does not describe how exactly we can find some structure in a graph, but only expect that weare able to do so. And this is the main direction for the future work. We should "translate" the algorithm to the language of computers, in other words, we should find out exact descriptions of structures and implement it. After this, we should be able to find all possible internally 4-connected counter-examples.

44 Bibliography

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