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Æ °±²³´Μ·¸¹º»¼½¾¿ Ý Masaryk University Faculty of Informatics Û¡¢£¤¥¦§¨ª«¬­Æ°±²³´µ·¸¹º»¼½¾¿Ý 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 graphf gG 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 incident2to 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 any2 pair of vertices2 x and^ y2of 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). ! 2 0 f g 2 0 Simple undirected graphs G; G are called isomorphic, written G ∼= G , if there exists a bijection # : V (G) V (G0), called an isomorphism, such that for each e = u; v , there exists an edge! #(u);#(v) and vice versa. Af graphg F results from G byf contractingg 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− formerlyf g [ f incidentg 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.
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