Masaryk University

Faculty of Informatics

Æ

Construction of planar emulators of graphs

Bachelor Thesis

Matˇej Klus´aˇcek

Brno, spring 2011 Declaration

Hereby I declare, that this paper is my original work, which I have elaborated on my own. All sources, references and literature used or excerpted during elab- oration of this work are properly cited and listed in complete reference to the appropriate source.

MatˇejKlus´aˇcek

Advisor: doc. RNDr. Petr Hlinˇen´yPh.D.

ii Acknowledgement

First of all I would like to express my gratitude to my advisor, doc. RNDr. Petr Hlinˇen´yPh.D., for his support, valuable and helpful remarks and suggestions. Moreover, I would like to thank Adam R˚uˇziˇcka for his comments and stylistic revision. My roommates deserve a special thanks for their patience during the time I was writing this thesis as well as my brother for checking some of my results. Finally, I am deeply grateful to my parents and my family for their both financial and moral support throughout my studies and whole my life.

iii Abstract

This thesis deals with the problem of describing the class of those graphs which have a planar emulator—i.e., a locally-surjective homomorphism from a planar graph. For a long time this class had been believed to coincide with the class of planar-coverable graphs (Fellows’ conjecture 1988), however Rieck and Yamashita showed the existence of nonprojective planar-emulable graphs in a recent breakthrough of 2008. In his Bc Thesis work, Derka studied the possible non-projective planar-emulable graphs further, and conjectured that all such ones are essentially internally-4-connected. Particularly, he conjectured that there are no planar emulators when it comes to graphs in the “K7 C4 family”. In this thesis we take the opposite standpoint, and construct− planar emulators of all members of the K C family—the four graphs among the 35 projective 7 − 4 forbidden minors of Archdeacon which are Y ∆-related to K7 C4. The common structural property of these graphs is the existence of an essential− 3-cut with both sides nonplanar which, at first sight, likely prevented a graph from having a planar emulator. We carefully describe the new emulators and their somehow odd behavior, and try to outline some properties they all share in order to make a further step towards understanding the mysterious class of graphs with planar emulators.

iv Keywords

Graph, finite planar emulator, 3-connected graph, projective plane, minor, Fellows’ conjecture

v Contents

1 Introduction ...... 1 2 Basic Definitions ...... 2 3 Planar cover and emulator problem ...... 5 4 Recent development in planar emulators ...... 11 5 Construction of new planar emulators ...... 14 5.1 Planar emulator of K C ...... 14 7 − 4 5.2 Planar emulator of 3 ...... 19 5.3 Planar emulator of D ...... 22 F1 5.4 Planar emulator of 5 ...... 22 6 Additional observationsE ...... 24 7 Conclusion and future research ...... 27 A Obstructions for the projective plane ...... i B Previously known emulators ...... ii C New emulators of graphs from K7 C4 family ...... vi D Adding new edges ...... − ...... x

vi 1 Introduction

Drawing graphs is the most intuitive way to represent them. Several points connected together with curves give the precise and complete characterization of a graph. The question of which graphs have planar embedding (i.e. which can be drawn without crossing lines) was therefore one of the most intriguing ones concerning the whole graph theory. Thanks to Kuratowski’s theorem, we can properly describe the whole class of them. However, what happens if we allow “multiplying” of vertices while preserving local structure of the graph? We say that a new graph has the “same local structure” if its vertices have the same neighbors as the vertices of the original graph. The only difference is in number of vertices - while every vertex is unique in the original graph, the new one can contain more copies of the same vertex. Such extended graphs are called planar covers and planar emulators, depending on the precise definition. Their most important property is that they do have planar embeddings even in cases when the original graph does not. Suddenly, we have a crucial question to consider: When does a planar cover or emulator exist? The problem was studied for the first time in the 1980s and two strong conjec- tures were introduced. Negami’s conjecture, which deals with the planar covers, has been almost proved [4, 5, 8, 9, 12] while Fellows’ one (concerning emulators) was after more than 20 years surprisingly falsified [13]! The breakthrough ba- sically showed that the class of planar-emulable graphs does not coincide with the planar-coverable one. In his Bachelor thesis [3], Martin Derka studied the possible planar-emulable graphs further, and proposed a new conjecture. This thesis focuses on specific graphs which were not believed, for several rea- sons, to have planar emulators. However, we took an opposite turn and tried to find at least some of them. Unexpectedly, we have finally succeeded in con- structing an emulator of the most important case and falsify even the most recent conjecture. By improving the approach we have successively described even more emulators. We have used some of the tools and ideas described by Fellows [5] and tried to outline some properties the new emulators share in order to make a further step towards understanding the class of graphs with planar emulators. This introductory part is followed by basic graph theory definitions which are necessary for the thesis. Chapter 3 introduces planar emulators in a more formal fashion presenting some of their important general properties and Chapter 4 provides a short summary of the recent development in this field. In Chapter 5 we present a detailed proof of existence of the previously unknown emulators together with their construction and Chapter 6 summarizes some of the properties they have in common. In Appendices, more illustrating pictures are presented in order to give a comprehensive view on the whole problem.

1 2 Basic Definitions

First of all, we present the core definitions necessary for the whole thesis. A simple undirected finite graph G is an ordered pair G = (V,E) where V is a set of vertices and E is a set of edges and both of them are finite. An edge e is an unordered pair of vertices e = u, v , sometimes simply referred to as e = uv. A set of vertices of a particular graph{ }G is denoted as V (G), a set of edges as E(G). Two vertices u and v of graph G are adjacent if there exists an edge uv E(G). Such vertices are often referred to as neighbors. The edge uv is incident∈to both u and v and is said to join or connect them. Vertices u and v are sometimes also called ends or endvertices of uv. Two edges are incident if they share a vertex. A walk in a graph is a sequence of adjacent vertices. The walk always has a start vertex and an end vertex. In a closed walk the start vertex equals to the end vertex, while in an open walk they differ. The length of a walk is the number of participating edges. A path in a graph is a walk, where no vertex is visited twice. A cycle in a graph is a path where the start vertex is the same as the end vertex. A graph is called connected if there exists a path between any two vertices and k-connected if it is connected, has at least k + 1 vertices and after removing at most k 1 vertices stays connected. A graph− H is a subgraph of a graph G if V (H) V (G), E(H) E(G) and for every edge uv E(H): u V (H) v V (H).⊆H is said to be⊆ an induced subgraph if, for any∈ pair of vertices∈ x and∧ y∈of H, xy is an edge of H if and only if xy is an edge of G. The degree of a vertex v in a graph G is the number of edges to which v is incident. The degree of v is denoted as degG(v). The vertex of degree 3 with distinct neighbours is called a cubic vertex. A graph homomorphism of H into G is a mapping h : V (H) V (G) such that, for every edge uv E(H), we have h(u), h(v) E(G). → ∈ 0 { } ∈ 0 Simple undirected graphs G, G are called isomorphic, written G ∼= G , if there exists a bijection ϑ : V (G) V (G0), called an isomorphism, such that for each e = u, v , there exists an edge→ ϑ(u), ϑ(v) and vice versa. A{ graph} F results from G by{ contracting} an edge e = uv if V (F ) = V (G) u, v w , where w is a new vertex, and E(F ) = E(G) e where all edges− {formerly} ∪ { incident} to u, v are now incident to w. Such a graph− is denoted by F = G/e. A graph H is called a minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge contractions on a subgraph of G. A subdivision H of a graph G is obtained by replacing some of the edges of G by new paths of an arbitrary length. A vertex separation (A, B) in a graph G is a pair of nonempty sets A, B such

2 2. Basic Definitions that A B = V (G) and there is no edge in G between sets A B and B A. The value∪ of A B is referred to as the order of separation (A,− B). − A graph G| is∩ said| to embed in a surface S if it is isomorphic to a drawing GS on S such that the points of GS representing the vertices are distinct and no two simple continuous curves representing the edges intersect each other. Such a drawing is also called an embedding.A face in an embedding is informally an empty region surrounded by a closed walk. The precise formal definition uses more advanced topological tools and is not necessary for our purposes. Vertices and edges which form this border walk are incident to this face and are referred to as its neighbors. Let F (G) be a set of faces of a graph G. A projective plane is the simplest non-orientable surface—a plane with one crosscap. The crosscap can be thought of as an object produced by puncturing a surface a single time, attaching two zips around the puncture in the same direction, distorting the hole so that the zips line up, requiring that the surface intersects itself, and then zipping up. The crosscap can also be described as a circular hole which, when entered, exits from its opposite point.

Fig. 1. An illustration of a crosscap in the picture on the left hand side and embeddings of the graphs K5 and K6 into the projective plane - the circle represents the crosscap on which the opposite points are identified.

A graph is called planar if it has an embedding in a plane, projective if it has an embedding in a projective plane and outer-planar if it has an embedding in a plane such that there exists a face incident to all the vertices. Theorem 2.1 (Euler’s formula). Let G be a planar graph. Then V (G) E(G) + F (G) = 2 | | − | | | | The class of planar and projective graphs can be characterized as follows: Theorem 2.2 (Kuratowski). A graph G is planar if and only if it does not contain a subdivision of K5 or K3,3 as a subgraph. Theorem 2.3 (Archdeacon [1]). A graph G is projective if and only if it does not contain one of the finite set of 35 graphs (Appendix A).

3 2. Basic Definitions

Fig. 2. An illustration of a Y ∆ transformation - replacing a cubic vertex by a triangle of its neighbors. The inverse operation is called ∆Y transformation

Let v G be a cubic vertex with distinct neighbours. Replacing v with a triangle on∈ its neighbours is called a Y ∆ transformation. The inverse operation (replacing a triangle face by new cubic vertex joined to all the vertices of the orig- inal triangle) is called a ∆Y transformation. The Y ∆ transformation preserves an embedding while the ∆Y transformation generally does not.

4 3 Planar cover and emulator problem

Informally, a planar graph H is an emulator of graph G if every vertex of G is represented by one or more vertices in H such that the following holds: Whenever two nodes v and u are adjacent in G, any node representing v in H has at least one (in case of an emulator) or exactly one (in case of a cover) adjacent node in H that represents u. Conversely, no node representing v in H has a neighbor representing u if v, u are nonadjacent in G. This chapter gives a short overview of the origin of planar cover and emulator problems as well as formal definitions and some well-known properties.

Definition 3.1. A finite planar graph H is a planar emulator (cover) of a graph G if there exists a graph homomorphism ϕ : V (H) V (G) such that, for every vertex v V (H), the neighbors of v in H are mapped→ by ϕ surjectively (bijectively) onto the∈ neighbors of ϕ(v) in G. The homomorphism ϕ is called an emulator (cover) projection.

The following two claims are immediately obtained:

Lemma 3.2. a) If H is a planar cover of G, then H is also a planar emulator of G. The converse is not true in general (Fig. 3). b) If G embeds in the projective plane, then G has a two-fold planar cover (i.e., ϕ−1(u) = 2 for all u V (G)); cf. [10]. See also Fig. 4. | | ∈

b6 c5 b2 a2 a b s s s s a3 b5 s s s s c2 c1 b3 c4 s s s s c3 b4 c s s s a1 b1 s s as4

Fig. 3. Triangle G = K3 (left) and examples of a planar cover (center) and a planar emulator (right). We simply denote the vertices representing a of G by ai, i = 1, 2,... , and with b, c analogically.

Concepts of planar covers and emulators emerged independently from works of Fellows [4, 5] (emulator) and Negami [10–12] (cover). The class of planar- coverable graphs seems to be better understood since at least we have the follow- ing nice topological characterization with rather trivial “if” direction:

Conjecture 3.3 (Negami [11], 1988). A graph has a (finite) planar cover if and only if it embeds in the projective plane.

5 3. Planar cover and emulator problem

c2 d2 ϕ(v1)= ϕ(v2)= v s s v2 c d b2 s a2 s s s s

d1 c1 v ←− s s s b a ϕ vs1 s s G = K5 a1 b1 z s s Fig. 4. The graph G = K5 (left) and its two-fold planar cover (right) via a homomorphism ϕ. The cover is obtained from a “crosscap-less” drawing of G and its mirror image.

This Conjecture 3.3 identifies the property of having a planar cover with the property of being projective. If a connected graph G is not projective, then G contains one of the projective forbidden minors (see Appendix A). The list of them is complete [1]. Hence, to prove Conjecture 3.3, it remains to show that the known 32 connected projective forbidden minors have no planar covers. However, this task has not been completed yet despite more than 20 years of intensive research (see [8]). Through a series of papers the following was established:

Theorem 3.4 (Archdeacon, Fellows, Hlinˇen´y,and Negami, 1988–98). If the graph K1,2,2,2 has no planar cover, then Conjecture 3.3 is true. The question of characterizing planar-emulable graphs was opened for 20 years as well, but apparently no significant effort was made to contradict the natural feeling that having locally more than one neighbor in H representing the same adjacent pair of G could ever help to gain planarity of H. Such a possibility seemed so highly counterintuitive that it was widely believed:

Conjecture 3.5 (Fellows [5], 1988, falsified 2008). A graph has a planar emulator if and only if (cf. Conj. 3.3) it embeds in the projective plane.

Applying the arguments which worked for planar covers to planar emulators, i.e. to Conjecture 3.5, seemed very natural indeed. The first positive partial result of Fellows [5] covered first 19 graphs in Appendix A at at the same time (Theorem 3.7). We say that a graph G contains two disjoint k-graphs if there exist two vertex- disjoint subgraphs J ,J G such that, for i = 1, 2, the graph Ji is isomorphic 1 2 ⊆ to a subdivision of K or K , , the subgraph G V (Ji) is connected and adjacent 4 2 3 − to Ji, and contracting in G all the vertices of V (G) V (Ji) into one results in a \ nonplanar graph (i.e. containing a K5- or K3,3-subdivision).

6 3. Planar cover and emulator problem

Fig. 5. An example of a graph having two disjoint k-graphs (shaded in gray).

The following lemma determines the lower bound for the size of emulators and is important for the proof of the further Theorem 3.7:

Lemma 3.6 (Fellows [5]). In every planar emulator H of a nonplanar con- nected graph G with the projection ϕ : V (H) V (G), the following holds: ϕ−1(v) 2 for each v V (G). → | | ≥ ∈ The proof can be found in [5].

Theorem 3.7 (Fellows [5]). A planar-emulable graph cannot contain two disjoint k-graphs. Consequently, each of the 19 graphs—projective forbidden minors—in the first three rows of Fig. 24 in Appendix A has no planar emu- lator.

Proof. Suppose, for a contradiction, that G contains two disjoint k-graphs (see an example in Fig. 5) J ,J G, and that there exists a planar emulator H with a 1 2 ⊆ projection ϕ : V (H) V (G). Let Hi, i = 1, 2, denote the subgraph of H induced → by the edges representing E(Ji) in the projection ϕ. (An edge f = xy E(H) ∈ represents e if e = ϕ(x), ϕ(y) .) Then H1 and H2 are vertex-disjoint, and up to symmetry between{H ,H , there} exists a component A H such that all other 1 2 1 ⊆ 1 components of H1,H2 lie in the outer face of A1 in the plane drawing of H. Since G V (J1) is connected and adjacent to J1, it follows that all the vertices of V (H) V−(A ) lie in the outer face of A . So, by contracting V (H) V (A ) into \ 1 1 \ 1 one vertex x we obtain a planar graph H0 which is an emulator of the nonplanar graph G0 resulting from G by contracting all V (G) V (J1) into one vertex w. Let \ −1 ϕ0 : V (H0) V (G0) be the derived emulator projection. Then ϕ0 (w) = x , which is a contradiction→ to the previous Lemma 3.6. { } ut

The nonexistence of another non-projective graph (K3,5) was consequently proved (the precise origin of the proof is unclear), but all the other attempts using similar tools as in the planar covers case failed apparently due to rather technical reasons. Therefore the Fellow’s conjecture was widely believed to be true.

7 3. Planar cover and emulator problem

Except for direct proofs concerning existence or nonexistence of certain emu- lators, a few interesting properties of the whole class of planar-emulable graphs were discovered. Most of them appeared already in [5], which has never been published. For the sake of completeness, we repeat the important ones here, as taken from [2] and [5].

Proposition 3.8 (Fellows [5]). The property of being planar-emulable is closed under taking minors.

Proof. Let G be a planar-emulable graph, and a planar graph H be its emulator via a projection ϕ. We prove this easy proposition by showing how H is modified to accommodate for the elementary reduction steps in G; vertex/edge deletion, and edge contraction. Say, if a vertex v V (G) is deleted, then also all vertices ϕ−1(v) representing v are deleted from H∈. An edge f = xy E(H) represents the edge e E(G) if e = ϕ(x), ϕ(y) . Whenever an edge e ∈ E(G) is deleted, so are all the∈ edges representing{ e in H}. Lastly, if an edge e ∈ E(G) is contracted, then every component induced by the edges representing∈ e in H is also contracted into a single vertex (note that such components may contain more than one edge representing e in the case of an emulator), and possible parallel edges are simplified. All these operations preserve planarity of H, and the outcome is an emulator of the graph resulting from G. ut Proposition 3.9 (Fellows [5]). The property of being planar-emulable is closed under applying Y∆-transformations (Figure 2).

Proof. Let G be a planar-emulable graph and v V (G) a vertex of degree 3. Denote by G0 the graph obtained from G by applying∈ the Y∆-transformation of v. Suppose a planar graph H that is an emulator of G via a projection ϕ. In the (optimistic) case that all the vertices of H in ϕ−1(v) are also of degree 3, we simply successively apply Y∆-transformations to all the vertices in ϕ−1(v) (which form an independent set of H), and the resulting graph H0 will be again planar and an emulator of G0. It remains to justify our optimistic assumption about degree-3 vertices in ϕ−1(v) of a suitable planar emulator H of G, which follows from the following claim applied to X = v : { } Lemma 3.10 (Fellows [5]). Let G be a planar-emulable graph and X V (G) an independent set of vertices of degree 3. Then there exists a planar emulator⊆ H of G with a projection ϕ : V (H) V (G) such that every vertex u ϕ−1(v) over all v X is of degree 3. → ∈ ∈

8 3. Planar cover and emulator problem

−1 −1 y1 ∈ ϕ (a) y1 ∈ ϕ (a) −1 −1 y2 ∈ ϕ (b) y2 ∈ ϕ (b)

−1 −1 y3 ∈ ϕ (a) y3 ∈ ϕ (a) x x2 →

x1

yi+1 yi+1 yi−1 yi−1 −1 −1 yi ∈ ϕ (c) yi ∈ ϕ (c)

Fig. 6. Splitting a vertex x with a cubic image into vertices of lower degree.

−1 −1 y1 ∈ ϕ (a) y1 ∈ ϕ (a) −1 −1 y2 ∈ ϕ (b) y2 ∈ ϕ (b)

−1 x1 −1 y3 ∈ ϕ (c) y3 ∈ ϕ (c) x → x2

−1 −1 y6 ∈ ϕ (c) y6 ∈ ϕ (c) −1 −1 y4 ∈ ϕ (a) y4 ∈ ϕ (a) −1 −1 y5 ∈ ϕ (b) y5 ∈ ϕ (b)

Fig. 7. Illustration of the last case of the proof of Lemma 3.10.

Proof. Whenever F is an emulator of our graph G with a projection ψ : V (F ) V (G); let Dg(F )( 3) shortly denote the maximal F -degree of the vertices→ u ψ−1(v) over all≥ v X. We choose H as a planar emulator of G with projection∈ ϕ such that the∈ value Dg(H) is minimized. Assume, for a contradiction, that Dg(H) > 3, and choose any vertex x ϕ−1(v) where v X such that x is of H-degree Dg(H) = d > 3. Let a, b, c be∈ the three neighbors∈ of v in G. We denote by w the circular word of length d over the alphabet a, b, c formed of the letters ϕ(y1)ϕ(y2) . . . ϕ(yd), where y1, . . . , yd are the neighbors{ of}x in H in this cyclic order. Then, one of the following three cases, up to symmetry, occurs in w:

– w contains a subword aa: By merging the corresponding two vertices of H representing a into one, the degree of x drops to d 1. − – w contains a subword aba: Without loss of generality, it is ϕ(y1) = ϕ(y3) = a, ϕ(y ) = b, and ϕ(yi) = c for some 4 i d (to be a valid emulator of G). We 2 ≤ ≤ modify H by splitting a vertex x into x1, x2 with ϕ(x1) = ϕ(x2) = ϕ(x), so that x1 is adjacent to y2, y3, . . . , yi and x2 to yi, . . . , yd, y1, y2; see Fig. 6. Clearly, the

9 3. Planar cover and emulator problem

degrees of x1, x2 are now smaller than d. – w = (abc)+: Then H may be modified as shown in Fig 7, and the degrees of the newly created vertices drop down to 3.

In each of the cases it is easy to see that the obtained graph H0 is still a valid planar emulator of G, and that only the degrees of some neighbors of x in H could have gone up from H to H0. Hence, as X is an independent set, we can repeat the above construction for all the vertices x ϕ−1(X) (which form an independent set in H, too) of degree d, and in finitely∈ many steps obtain a contradiction to minimality of Dg(H). ut

10 4 Recent development in planar emulators

The purpose of this section is to give a review of the recent development in the field, because in 2008 the research took quite unexpectedly a different direction, in which this thesis continues—Fellow’s conjecture was surprisingly falsified:

Fig. 8. A planar emulator of K4,5 − 4K2

Theorem 4.1 (Rieck and Yamashita [13], 2008). The graphs K1,2,2,2 and K4,5 4K2 do have planar emulators (cf. Fig. 9). Consequently, the class of planar-emulable− graphs is strictly larger than the class of planar-coverable graphs, and Conjecture 3.5 is false.

We remark that this is not merely an existence result—the specific (and, sur- prisingly, not so large) emulators were published together with it (see Figures 8, 9 and Appendix B). Both K , , , and K , 4K are among the projective for- 1 2 2 2 4 5 − 2 bidden minors, and K4,5 4K2 has already been proved not to have a planar cover. − Furthermore, Theorem 4.1 is not a rarity at all. Many other nonprojective graphs have planar emulators as well.

Theorem 4.2 (Chimani and Hlinˇen´y). All of the graphs (Fig. 24 in Ap- pendix A) , , , , have planar emulators (see Appendix C). B7 C3 C4 D2 E2 11 4. Recent development in planar emulators

←

Fig. 9. An “artistic” colour-coded picture of a planar emulator (right) of the graph K4,5 − 4K2 (left), taken from http://vivaldi.ics.nara-wu.ac.jp/~yamasita/emulator/.

Inspired by these surprising results Martin Derka tried to characterize the class of planar-emulable graphs in his bachelor thesis [3]. He used a similar approach as Hlinˇen´yand Thomas in [9] on the field of finite planar covers. Their main idea introduced in [9] was that if the last remaining case for deciding whether Conjecture 3.3 was true or not (K1,2,2,2) could not be solved directly, one might have a look on possible counterexamples. Hlinˇen´yand Thomas finally succeeded in generating a sporadic family of 16 graphs somehow derived from K1,2,2,2 itself and proved that every possible counterexample for Conjecture 3.3 must be a planar expansion of one of these graphs. The common feature (up to trivial modifications) of these graphs was the following connectivity property: A graph G is internally 4-connected if it is simple and 3-connected, it has at least 5 vertices and for every separation (A, B) of order 3, either G  A or G  B has at most 3 edges. In other words, it is important whether our graph has a “substantial” 3-cut. Derka uses similar tools together with computerized power and generates a lot of non-projective graphs which might be other counterexamples to Fellow’s conjecture. During this generating process new graphs are either refused (if they contain a minor not believed to have a planar emulator) or accepted (otherwise). A lot of graphs resulting from K1,2,2,2, 7, 3, 4, 2, 2 which might have a finite planar emulator were found and the computationsB C C D haveE not finished yet. Since each of the graphs from K7 C4, 3, 5, 1 (often denoted as K7 C4 family) has a “substantial’ ’ 3-cut{ and− thereforeD E isF internally} 4-connected (unlike− the rest of graphs with previously known planar emulators), and covers and emulators were still believed to be two sides of the same coin, Derka refused also all the generated graphs which have a minor among them. Finally, he proposed a new conjecture:

Conjecture 4.3 (Derka [3], 2010, falsified 2010). There exists a finite set Γ of internally 4-connected non-projective graphs that have planar emulators.

12 4. Recent development in planar emulators

A graph G has a planar emulator if and only if G embeds in the projective plane or G is a planar expansion of some graph H from Γ .

Obviously, this can hold only if none of the graphs from the K7 C4 family has a planar emulator. −

13 5 Construction of new planar emulators

Our primary research objective was to investigate the graphs in K7 C4 family (see Figure 10) and prove the existence or non-existence of planar emulators− of them. These graphs are among the projective forbidden minors (Appendix A) and were not believed to have emulators since they are not internally 4-connected. Therefore answering the question whether any of them has a planar emulator was the next important step in characterizing the whole class of planar-emulable graphs. We tried to take an opposite standpoint than the others and unexpectedly succeeded in constructing new planar emulators of all of them, which is the main result of this thesis.

Fig. 10. Graphs from the K7 − C4 family: {K7 − C4, D3, E5, F1}. The vertices and edges from a “substantial” 3-cut are shaded in grey.

Therefore, we state a new theorem:

Theorem 5.1. The graphs K7 C4, 3, 5, 1 (Figure 10) do have a finite planar emulator. { − D E F } It gives us, together with Theorems 4.1 and 4.2, planar emulators of all the pro- jective forbidden minors—except K4,4 e perhaps— that have been in doubt since Fellows’ [5]. Simultaneously it falsifies− the previous Conjecture 4.3. Consequently, the class of planar-emulable graphs is, in fact, much larger and structurally richer than the class of planar-coverable ones, and therefore the concept of projective embeddability seems very out-of-place in the context of planar emulators. We prove our Theorem 5.1 by giving the description of the process of “build- ing” these emulators and try to analyze their common properties and possible extensions of them which have emulators as well.

5.1 Planar emulator of K7 − C4

The existence of a planar emulator of K7 C4 can be proved easily by giving its description or illustration directly. However,− constructing such an emulator follows a sequence of logic steps and cannot be done randomly. Therefore, we present a more detailed proof including the ideas behind. Firstly, we introduce this definition:

14 5. Construction of new planar emulators

Definition 5.2. Let G be a graph we want to find an emulator of, H an arbitrary graph and ϕ a mapping V (H) V (G). A vertex a ϕ−1(v) of graph H is satisfied if it has all the neighbors→ the original vertex v∈has { in G.}

Since K7 C4 is 3-connected and a cycle consisting of these 3 vertices divides the graph into− two identical components, it might be useful to divide the vertex set of K7 C4 into three groups: the triple of central vertices (named 1, 2, 3 on the very left− in Figure 10) adjacent to all other vertices, and the two vertex pairs (named A, B and C,D) each of which has connections only to its “mate” and to the central triple. Now we prove the following lemma:

Lemma 5.3. Let H be an emulator of K7 C4. Then we can always add (if necessary) some edges to H, such that no vertex− x A, B is incident to the same face as any vertex y C,D . In other words,∈ we { can modify} the emulator in such a way that every∈ connected { } subgraph in H on vertices from ϕ−1(A) ϕ−1(B) lies in a face bordered by a cycle of vertices from ϕ−1(1) { ϕ−1(2) ∪ ϕ−1(3) }. The same holds for connected subgraphs on vertices{ from ∪ϕ−1(C) ∪ ϕ−1(D)} . { ∪ }

Fig. 11. The top-left picture shows a subgraph S and its neighbors. The only possible obstruction for forming a cycle bordering the subgraph S from Lemma 5.3 is shown in the top-right picture and the remaining two drawings present the two possible cases which can occur while solving this situation. In both of them we arrive to a contradiction.

15 5. Construction of new planar emulators

Proof. Consider all the neighbors of such a subgraph S and denote the set of them as N. According to the property of planar emulators, only images of central vertices can be in N (see Figure 11). Since central vertices are mutually adjacent in K7 C4, we can add edges between each two vertices from N which are “close” to each− other on the planar drawing of H. By being “close” we mean that there exist a path from one to the other, such that, in each visited vertex, it takes the first edge on the left (obviously it makes sense only on specific embeddings). If two vertices with the same label are “close” to each other, we connect them as well, and the edge can be later contracted without affecting the emulator. If a new edge crosses an edge from planar embedding of H, then the old edge must join – a vertex from S with an image of a central vertex (x). In that case we get a contradiction, because x is not in N although it is a neighbor of a vertex from S. – a vertex from S with an vertex from ϕ−1(C) ϕ−1(D) , which is a contradiction to the definition of planar emulators.{ ∪ } The same holds for the symmetric case of (C,D). Figure 11 shows the possible cases. Therefore every such subset S can be bordered by a cycle of vertices 1, 2, 3 . { } ut Definition 5.4. A skeleton of a planar emulator H of a k-connected graph G is an induced subgraph on vertices from set X, such that G remains disconnected after removing vertices ϕ(X).

In this case the smallest skeleton will be a subgraph induced on the ver- tices representing the central triple 1, 2, 3 and we place the remaining vertices representing A, B and C,D into the skeleton faces, provided certain additional requirements are met. These simple ideas lead to the introduction of basic building blocks each of which “almost” emulates the subgraph induced on 1, 2, 3, A, B and 1, 2, 3,C,D, respectively. These blocks must be obviously planar and we need the central vertices to be on the outer face in order to connect them into a skeleton later. We consider them forming a cycle according to Lemma 5.3. Vertices A, B (C,D) must be inside and must be satisfied. More formally, we are looking for a partial emulator of a complete subgraph (K5) of K7 C4 with central vertices on the outer face. The concept of a partial emulator is− inspired by the semi-cover firstly used by Hlinˇen´yin [7].

Definition 5.5. A finite planar graph H is a partial planar emulator of the graph G if there exists a graph homomorphism ϕ : V (H) V (G) such that, for every vertex v V (H) which is not incident to the outer→ face, the neighbors of v in H are mapped∈ by ϕ surjectively onto the neighbors of ϕ(v) in G.

16 5. Construction of new planar emulators

Fig. 12. All possible partial emulators (def. 5.5) of the complete subgraph (K5) of K7 − C4 with a hexagonal outer face which is incident to 3 central vertices only. Notice that most profitable cases (third and fourth ones) use splitting of an inner vertex.

Fig. 13. The most profitable building blocks for our K7 − C4 planar emulator and their arranging into a shape of an octahedron from an ”artistic” point of view.

If we take a triangle (cycle of length 3) as the outer cycle of the partial emu- lator, the inner vertices cannot be satisfied since K5 itself is not planar according to Kuratowski (Theorem 2.2). Therefore only the longer cycles can be taken into account. The hexagon (cycle of length 6) has a very nice property - if we take each central vertex twice, we can achieve the state, when all these vertices have all the required neighbors among themselves (see Figure 12). Now we investigate the possibilities of placing the inner vertices. The intuitive goal we want to achieve is the maximal number of different neighbors for each central vertex incident to the outer face. Figure 12 shows all the possible partial emulators of K5 (up to sym- metries) with a hexagonal outer face which is only incident to 3 central vertices. Notice that the most profitable cases use splitting an inner vertex which clearly shows how we can profit from the emulator concept compared to the cover one. Once these partial emulators have been found, we should ask how the skeleton might look like. Firstly, we pick building blocks no. 3 and 4 in figure 12, which have a very nice property - three central vertices on the outer face, none of which is adjacent to any other, have both of the inner vertices as neighbors (let them be called X,Y,Z). The remaining three (let them be called X,¯ Y,¯ Z¯) are joined to only one of them, on the other hand, this vertex can have an arbitrary label

17 5. Construction of new planar emulators

Fig. 14. A planar emulator of K7−C4, constructed from the blocks in Fig. 13. The skeleton representing the central vertices is drawn in bold. since blocks 3 and 4 in Figure 12 provide all possible combinations of the inner vertices (using rotations and switching the labels of two inner vertices). Therefore vertices (X,Y,Z) can be included only in two building blocks in the final emulator (among which one “almost” emulates 1, 2, 3, A, B and the other 1, 2, 3, A, B ) while the others need four. Now we look{ at the partial} emulators{ from another} point of view - consider them to be triangles with three vertices X,¯ Y,¯ Z¯ and three edges X,Y,Z . The problem of satisfying all the vertices and{ finding} an { } emulator of K7 C4 is equal to finding a planar graph with all the vertices of degree four, where− all faces are triangles and which is “face-2-colorable”. A graph G is face-k-colorable if there exists a surjective projection ϕ : F (G) 1, 2, ..., k such that if F , F are two incident faces, then ϕ(F ) = ϕ(F ). → { } 1 2 1 6 2 The smallest such graph is an octahedron (see Figure 13). Taking four partial emulators of 1, 2, 3, A, B and four of 1, 2, 3,C,D with proper rotation of the inner neighbors{ (which} is possible thanks{ to building} blocks no.3 and 4 in Figure 12) gives us the final planar emulator of K7 C4. The emulator has 120 edges on 42 vertices (see Figure 14) and already its existence− is quite unexpected.

18 5. Construction of new planar emulators

5.2 Planar emulator of D3

Once we can emulate K7 C4, the natural question to ask is if this construction can be extended to , a− graph which is created by applying a single ∆Y trans- D3 formation on K7 C4. First of all, we divide the vertex set of 3 into three groups as well: the triple− of central vertices (named 1, 2, 3), the vertexD pair (named E,F ) and the vertex triple (named A, B, C).

Lemma 5.6. Let G be a planar-emulable graph and X V (G) an independent ⊆ set of vertices of degree 3. Each of x X has neighbors rx, sx, tx . Then any ∈ { } of edges rxsx, sxtx, rxtx for every x can be added to G without preventing it from being planar-emulable.

Proof. The Lemma directly follows from Lemma 3.10. All the ϕ−1(x) are cubic also in an emulator so that every pair of its neighbors lie on the same face and therefore can be joined by a new edge. ut

Fig. 15. All possible partial emulators 5.5 of a induced subgraph of D3 on vertices {1, 2, 3, A, B, C} using a cycle of length 6 consisting only of central vertices 1, 2, 3. Notice that the best properties can be achieved again by splitting an inner vertex (right-most picture).

Lemma 5.6 allows us to add an edge 23 into 3 because vertex B (for example) has degree 3 and both vertices 2 and 3 are its neighbors.D Now the same argument concerning a skeleton of an emulator as for K7 C4 can be applied. Again, the smallest skeleton is a subgraph induced on the− vertices representing the central triple 1, 2, 3 and we place the remaining vertices representing A, B, C and E,F into the skeleton faces. Obviously applying the identical approach as in the previous case does not work because of a special property of vertex 1. If we take the same partial emulators as in K7 C4 case and upgrade one of them in order to “almost” emulate 1, 2, 3, A, B,− C (see the picture on the very right in Figure 15), the vertex labeled 1 has either one or no neighbor labeled A. Therefore we cannot use an

19 5. Construction of new planar emulators octahedron, since two vertices labeled 1 would have no neighbor A (we have six vertices labeled 1 and only four faces covered with partial emulators of the proper part of 3). Nevertheless,D we can take a half of this construction as a “core” and “fix” the properties of vertices causing obstructions.

Fig. 16. The supporting blocks for D3’s emulator.

Fig. 17. The construction derived from one half of the emulator of K7 − C4 and 8 small cells for the outer vertices to have the maximal number of different neighbors

We define two other supporting blocks, (see Figure 16). These will help us overcome the drawback. We take the aforementioned core and surround the graph

20 5. Construction of new planar emulators

Fig. 18. The finite planar emulator of D3

21 5. Construction of new planar emulators with 8 new blocks, such that the central vertices have all the desired neighbors among 1, 2, 3 and the outer vertices of the new expanded graph have better properties{ concerning} the number and kind of neighbors (see Figure 17). More precisely, if any one of these vertices misses a neighbor from A, B, C , then it has all the desired neighbors among E,F and vice versa. This{ fact enables} us to make a copy of the graph and join{ it in} a smart way using different simpler building blocks with the original graph to obtain an emulator. The final emulator of 3 (consisting of 280 edges on 122 vertices) is shown in Figure 18 and the approach,D described above, is clearly visible (two identical “cores” derived from an emulator of K C connected together). 7 − 4

5.3 Planar emulator of F1 The construction of an emulator of follows the same pattern as building an F1 emulator of 3. In fact, the emulator of 1 (with 256 edges on 138 vertices) was found firstD by the above mentioned constructionF and the emulator of D3 resulted from a simplification of an emulator of 1,( 3 results from 1 by taking a “Y ∆ transformation”, which is trivial to performF inD the emulator)F according to Proposition 3.9.

5.4 Planar emulator of E5

Fig. 19. Building cells for E5’s emulator

In order to obtain an emulator of 5, we again take the same “core” as in previous cases. Similarly, we use someE smaller additional cells to improve the properties of the outer vertices of the core (see Figure 19). Since is slightly E5 22 5. Construction of new planar emulators

Fig. 20. A construction built of a half of the K7 − C4’s emulator with upgraded partial emulators and 8 additional cells such that outer vertices have “nice” properties concerning the number and type of theirs neighbors.

different from 1—both come from K7 C4, but two ∆Y transformations took place in differentF triangle faces, the use− of the supporting small cells is quite different as well. We surround the core as showed in Figure 20. In this way we get better properties of the outer vertices of the new graph. Outer vertices labeled 1 have all desired inner neighbors (labeled with letters), vertices 2 miss one neighbor among A, B, C and one among D,E,F and vertices labeled 3 miss only one neighbor{ among} A, B, C or D,E,F{ . } Now we use a similar tool{ to the one} used{ in the} previous cases of and D3 1– we duplicate the graph and connect the two copies in a clever way so that verticesF labeled 1 get the desired neighbors labeled 2 and four new hexagons are created. The vertices surrounding each hexagonal face lack some neighbors of the same component. We fill each face with a proper building block (see Figure 19) to satisfy all the remaining vertices. Now we have a complete emulator of 5 with 272 edges on 138 vertices (see Figure 34 in Appendix C). E By giving the constructions of these four emulators we prove our main Theo- rem 5.1. ut

23 6 Additional observations

First of all, we can have a look at the edges and vertices which can be possibly added to graphs from K7 C4 family and do not prevent the graph from having a planar emulator. According− to Lemma 5.6 quite lot of edges can be added. Figure 36 shows the case of , the others can be found in Appendix D. E5

Fig. 21. Edges which can be added to the graph E5 without affecting the property of having a planar emulator.

Many other planar-emulable graphs can be obtained by applying planar ex- pansions. A planar expansion of a graph G is a graph which results from G by adding a planar graph sharing one vertex with G, or by replacing an edge or a cubic vertex with a planar graph with its attachments on the outer face. When we take into account all the known emulators of non-projective minor- minimal graphs, some regularities can be found. For example, some of them can be easily and beautifully embedded on polyhedrons (see Figure 22). K C 7 − 4 can be embedded on an octahedron, K , 4K and K , , , on a cuboctahedron, 4 5 − 2 1 2 2 2 2 on a truncated cube... However, it is probably only a result of the symmetric structureE of the graphs they are emulating.

Fig. 22. Some emulators can be nicely embedded onto polyhedrons - K7 − C4 onto an octahedron (left) and K4,5 − 4K2 onto a cuboctahedron(right).

The number of vertices of a minimal planar cover is bounded by a constant multiple of the number of vertices of the original non-planar graph [9]. When we look at the size of the emulators, we obtain the following numbers:

24 6. Additional observations

– K1,2,2,2 is a graph with 18 edges on 7 vertices and its smallest known emulator has 120 edges on 42 vertices.

– 7 is a graph with 18 edges on 8 vertices and its smallest known emulator has 120B edges on 48 vertices.

– 3 is a graph with 18 edges on 9 vertices and its smallest known emulator has 120C edges on 54 vertices.

– 4 is a graph with 18 edges on 9 vertices and its smallest known emulator has 744C edges on 338 vertices.

– 2 is a graph with 18 edges on 10 vertices and its smallest known emulator has 120D edges on 60 vertices.

– K4,5 4K2 is a graph with 16 edges on 9 vertices and its smallest known emulator has 96− edges on 50 vertices.

– K7 C4 is a graph with 17 edges on 7 vertices and its smallest known emulator has− 120 edges on 42 vertices.

– 3 is a graph with 16 edges on 8 vertices and its smallest known emulator has 280D edges on 122 vertices.

– 5 is a graph with 16 edges on 9 vertices and its smallest known emulator has 272E edges on 138 vertices.

– 1 is a graph with 15 edges on 9 vertices and its smallest known emulator has 256F edges on 138 vertices.

The number of vertices of a minimal planar emulator of a non-projective graph seems to be at least the sextuple of the number of vertices of the original graph (according to known emulators of K1,2,2,2 and K7 C4 families. On the other hand, some of the emulators are noticeably large. However,− we still do not have any tool to prove lower or upper bound of the number of vertices. ∆Y transformations performed on original graphs sometimes noticeably in- crease the complexity and size of an emulator (confront K C and or 7 − 4 D3 B7 and 4) and sometimes the emulators are very similar ( 7 and 3). However, tak- ing ∆YC transformations does not preserve the propertyB of beingC planar-emulable. The fact was mentioned in [2] and we give it as a proper Lemma:

Lemma 6.1. The property of having a finite planar emulator is not closed under taking ∆Y transformations.

Proof. Let us consider the graph 3 and label the vertices as in Fig. 23. Now we delete the edge 3, 8 and getB a subgraph which is projective (minors form Figure 24 are minimal{ non-projective)} and therefore has a planar cover (and an emulator as well). Now we perform a single ∆Y transformation on a triangular face 567 and obtain the graph 4, which is on the list of forbidden minors for the projective plane. Hence, theD property of having a planar emulator cannot be closed under performing ∆Y transformations. ut 25 6. Additional observations

Fig. 23. The process of creating non-planar-emulable D4 (right bottom) from the strict subgraph of B3 obtained by removing one edge (left top) and applying a single ∆Y transformation on 567 triangle face.

Not all of the emulators have the number of vertices equal to a multiple of the number of vertices of the original graph (K4,5 4K2, 3...) The question is whether it is just an anomaly and these emulators− can beD simplified or if the emulators with a “nice” number of vertices are a coincidence. Another interesting thing is the similarity between an emulator of K C 7 − 4 and an emulator of K1,2,2,2. Those two graphs are the smallest non-projective subgraphs of K7 and their emulators have exactly the same number of vertices and edges. Is it only a coincidence or is there any structural dependency which determines these two numbers to be the same?

26 7 Conclusion and future research

The main goal of this thesis was achieved—we proved the existence of a planar emulator not only for K7 C4, but also for the rest of the graphs from the whole K C family. However,− the results are so surprising that they bring a lot of new 7 − 4 questions as well. The existence of an emulator of K7 C4 forces us to reformulate conjectures and extend the research to more graphs− than only non-projective minors. The main and crucial question of finding a general characterization of the class of planar-emulable graphs still remains unsolved. Instead, we suggest to consider the following specific subproblems for the future research: – Is there a planar emulator of the last remaining case among the list of minor- minimal non-projective graphs (K4,4 e)? In our opinion the answer is no for several reasons. Anyway, we are currently− unable to find a proof, e.g. extending the arguments of [6]. – What impact do the so-called ∆Y transformations have on the property of having planar emulator? By Lemma 6.1 the class of planar-emulable graphs is not closed under taking these transformations, on the other hand, sometimes we can quite easily cope with it.

– Are there smaller emulators of 3, 5, 1 and 4? If the currently known emulators are minimal, then the complexityD E F of theC whole problem may be even higher than we believe. – The two smallest projective forbidden minors which are subgraphs of K (K C 7 7− 4 and K1,2,2,2) have planar emulators. What is the minimal subgraph of K7 which does not have a planar emulator? – The size of a minimal planar cover is bounded by a constant multiple (2 by Conjecture 3.3) of the number of vertices of the original graph [9]. Can anything× similar hold (be conjectured) for emulators?

27 References

1. D. Archdeacon, A Kuratowski Theorem for the Projective Plane, J. Graph Theory 5 (1981), 243–246. 2. M. Chimani, M. Derka, P. Hlinˇen´y,M. Klus´aˇcek, How Not to Characterize Planar-emulable Graphs, submitted, 27 p. 3. M. Derka, Planar Graph Emulators: Fellows’ Conjecture, Pr´ace SVOC,ˇ Masaryk University, Brno, 2010. 4. M. Fellows, Encoding Graphs in Graphs, Ph.D. Dissertation, Univ. of Califor- nia, San Diego, 1985. 5. M. Fellows, Planar Emulators and Planar Covers, Unpublished manuscript, 1988. 6. P. Hlinˇen´y, K4,4 e Has No Finite Planar Cover, J. Graph Theory 27 (1998), 51–60. − 7. P. Hlinˇen´y, Another Two Graphs Having no Planar Covers, J. Graph Theory 37 (2001), 227–242. 8. P. Hlinˇen´y, 20 Years of Negami’s Planar Cover Conjecture, Graphs and Com- binatorics 26 (2010), 525–536. 9. P. Hlinˇen´y,R. Thomas, On possible counterexamples to Negami’s planar cover conjecture, J. of Graph Theory 46 (2004), 183–206. 10. S. Negami, Enumeration of Projective-planar Embeddings of Graphs, Discrete Math. 62 (1986), 299–306. 11. S. Negami, The Spherical Genus and Virtually Planar Graphs, Discrete Math. 70 (1988), 159–168. 12. S. Negami, Graphs Which Have No Finite Planar Covering, Bull. of the Inst. of Math. Academia Sinica 16 (1988), 378–384. 13. Y. Rieck, Y. Yamashita, Finite planar emulators for K4,5 4K2 and K1,2,2,2 and Fellows’ Conjecture, European Journal of Combinatorics− 31 (2010), 903– 907.

28 Appendix A Obstructions for the projective plane

K K K K K K 3,3 · 3,3 5 · 3,3 5 · 5 B3 C2 C7

D1 D4 D9 D12 D17 E 6 E11

E19 E20 E27 F 4 F 6 G1

K K 4K K e K C 3,5 4,5 − 2 4,4 − 7 − 4 D3 E5 F 1

K 1,2,2,2 B7 C3 C4 D2 E 2

Fig. 24. The 32 connected projective forbidden minors. (The three disconnected ones, K5 + K5, K5 + K3,3, K3,3 + K3,3, are skipped since they clearly have emulators.)

i Appendix B Previously known emulators

Fig. 25. The finite planar emulator of K1,2,2,2

Fig. 26. The finite planar emulator of B7

ii B. Previously known emulators

Fig. 27. The finite planar emulator of C3

Fig. 28. The finite planar emulator of D2

iii B. Previously known emulators

Fig. 29. The finite planar emulator of C4

iv B. Previously known emulators

Fig. 30. The finite planar emulator of E2

v Appendix C New emulators of graphs from K C family 7 − 4

Fig. 31. The finite planar emulator of K7 − C4

vi C. New emulators of graphs from K C family 7 − 4

Fig. 32. The finite planar emulator of D3

vii C. New emulators of graphs from K C family 7 − 4

Fig. 33. The finite planar emulator of F1

viii C. New emulators of graphs from K C family 7 − 4

Fig. 34. The finite planar emulator of E5

ix Appendix D Adding new edges

Fig. 35. Edges which can be added to the graph D3 without affecting the property of having a planar emulator.

Fig. 36. Edges which can be added to the graph E5 without affecting the property of having a planar emulator.

Fig. 37. Edges which can be added to the graph F1 without affecting the property of having a planar emulator.

x