Peter Zeman. Algebraic, Structural and Complexity Aspects of Geometric Representations of Graphs

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1 MASTER THESIS Peter Zeman Algebraic, Structural and Complexity Aspects of Geometric Representations of Graphs Computer Science Institute Supervisor of master thesis: Study program: Study branch: RNDr. Pavel Klavík Computer Science Discrete Models and Algorithms Prague 2016

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3 Acknowledgements The results presented in this thesis are based on: [52] Pavel Klavík and Peter Zeman. Automorphism Groups of Geometrically Represented Graphs. In Leibniz International Proceedings in Informatics (LIPIcs), 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). 30: [53] Pavel Klavík and Peter Zeman. Automorphism Groups of Geometrically Represented Graphs (full version). Submitted, 2016, pre-print: [14] Steven Chaplick, Martin Töpfer, Jan Voborník, and Peter Zeman. On H-Topological Intersection Graphs. Submitted, First of all I would like to thank RNDr. Pavel Klavík for supervising my master thesis. We studied symmetries of interval graph in my bachelor thesis. The main result is that the automorphism groups of interval graphs are exactly the same as the automorphism groups of trees. Later, we considered this problem for circle graphs. Based on those results we wrote the paper [52]. In this paper we also propose a general technique which can be applied to other classes. Using this technique we were also able to give a characterization of automorphism groups of comparability and permutation graphs [52]. All those results are covered in Chapters 3, 4, and 5. A significant part of Chapter 6 (Sections 6.1.1, and Section 6.2.2) was done in collaboration with Martin Töpfer and Jan Voborník during a stay at the Rutgers University as a part of Research Experiences for Undergraduates The remaining parts of this chapter were discussed with Steven Chaplick, a postdoctoral researcher at University of Würzburg. This collaboration was started at the workshop GROW I would like to thank all the members of Computer Science Institute and Department of Applied Mathematics for the support and a nice environment. I am also grateful to my family, especially my parents, who supported me during my studies. 3

4 I declare that I carried out this master thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act. Prague, July 27, 2016 Peter Zeman 4

5 Contents 1 Introduction: Geometric Representations of Graphs Symmetries of Geometrically Represented Graphs Interval Graphs Circle Graphs Comparability Graphs Related Graph Classes Automorphism Groups Acting on Intersection Representations Algorithmic and Complexity Results for H-graphs Related Graph Classes Our Results Preliminaries: Group Theory Group Products Direct Product Semidirect and Wreath Products Automorphism Groups of Disconnected Graphs Automorphism Groups of Trees Automorphism Groups of Pseudotrees Interval Graphs PQ- and MPQ-trees PQ-trees MPQ-trees Automorphisms of MPQ-trees Automorphism Groups of Interval Graphs The Action on Interval Representations

6 3.3.2 Direct Constructions Automorphism Groups of Unit Interval Graphs Circle Graphs Split Decomposition Split Tree Automorphisms of Split Trees Recursive Construction of The Automorphism Group The Action On Prime Circle Representations Automorphism Groups of Circle Graphs The Action on Circle Representations Comparability and Permutation Graphs Modular Decomposition Modular Tree Automorphisms of Modular Trees Recursive Construction of The Automorphism Group Automorphism Groups of Interval Graphs Automorphism Groups of Comparability Graphs Structure of Transitive Orientations Action Induced On Transitive Orientations Automorphism Groups of Permutation Graphs Action Induced On Pairs of Transitive Orientations The Inductive Characterization Linear-time Algorithm Automorphism Groups of Bipartite Permutation Graphs k-dimensional Comparability Graphs H-graphs Recognition S d -graphs T -graphs Bounded List Coloring of Co-comparability Graphs Hardness for Diamond-Graphs Dominating Set

7 6.2.1 Interval Graphs S d -graphs H-graphs Graph Isomorphism and Maximum Clique Maximum Clique in Helly H-graphs Hardness Results Coloring and Treewidth Conclusions Open Problems Circular-Arc Graphs A Dichotomy on H-graphs

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9 Nazov: Autor: Katedra: Veduci prace: veduceho: Kl'ucove slova: Abstrakt: Algebraické, štrukturálne a výpočetné vlastnosti geometrických reprezentacií grafov Peter Zeman Informatický ústav Univerzity Karlovy RNDr. Pavel Klavík klavik@iuuk.mff.cuni.cz grupy automorfizmov, intervalové grafy, circle grafy, comparability grafy, H-grafy, rozpoznávanie, dominujúca množina, grafový izomorfizmus, maximálna klika, farbenie V tejto práci študujeme symetrie geometricky reprezentovatelných grafov. Popisujeme techniku, pomocou ktorej je možné určit grupu automorfizmov grafu tak, že analyzujeme jej akciu indukovanú na množine možných geometrických reprezentácií. Ukazujeme, že grupy automorfizmov intervalových grafov sa zhodujú s grupami automorfizmov stromov. Navyše pre daný intervalový graf skonštruujeme strom s rovnakou grupou automorfizmov. Toto rieši otázku Hanlona [Trans. Amer. Math. Soc 272(2), 1982]. Ďalej charakterizujeme grupy automorfizmov permutačných a circle grafov pomocou grupových súčinov. Taktiež ukazujeme, že každá abstraktná grupa sa dá realizovat pomocou čiastočného usporiadania dimenzie najviac štyri. Ďalšou z hlavných tém práce sú H-grafy, ktoré zaviedli Biró, Hujter, a Tuza v roku Sú to prienikové grafy súvislých podgrafov podrozdelenia nejakého zafixovaného grafu H. Táto práca je prvé štúdium H-grafov z hl adiska významých problémov teoretickej informatiky. Ukazujeme, že rozpoznávanie H-grafov je NP-úplné ak H je diamant (úplný graf na štyroch vrcholoch bez hrany). Toto rieši otázku Birá, Hujtera a Tuzy, že či je možné rozpoznávat H-grafy v polynomiálnom čase pre fixné H. Ďalej ukazujeme, že rozpoznávanie H-grafov je polynomiálne ak H je strom a, že je možné rozpoznávat S d -grafy v cǎse O(n 4 ), kde S d je úplný bipartitný graf K 1,t. Ukazujeme, že grafový izomorfizmus je GI-úplný a hl adanie maximálnej kliky je APX-t ažké ak H obsahuje určitý multigraf ako minor. Konečne dokazujeme, že hl adanie najmenšej dominujúcej mnoźiny je možné riešit v FPT pre S d -grafy a v XP pre H-grafy. 9

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11 Title: Author: Department: Supervisor: Supervisor's Keywords: Abstract: Algebraic, Structural and Complexity Aspects of Geometric Representations of Graphs Peter Zeman Computer Science Institute RNDr. Pavel Klavík automorphism groups, interval graphs, circle graphs, comparability graphs, H-graphs, recognition, dominating set, graph isomorphism, maximum clique, coloring We study symmetries of geometrically represented graphs. We describe a technique to determine the automorphism group of a geometrically represented graph, by understanding the structure of the induced action on all geometric representations. We prove that interval graphs have the same automorphism groups as trees, and for a given interval graph, we construct a tree with the same automorphism group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982]. For permutation and circle graphs, we give an inductive characterization by semidirect and wreath products. We also prove that every abstract group can be realized by the automorphism group of a comparability graph/poset of the dimension at most four. We also study H-graphs, introduced by Biró, Hujter, and Tuza in Those are intersection graphs of connected subgraphs of a subdivision of a graph H. This thesis is the first comprehensive study of the complexity of several important computational problems in theoretical computer science on H-graphs. We negatively answer the question of Biró, Hujter, and Tuza who asked whether H-graphs can be recognized in polynomial time, for a fixed graph H: it is NP-complete when H is the diamond graph. For each tree T, we give a polynomial-time algorithm for recognizing T -graphs and an O(n 4 )-time algorithm for recognizing star-graphs, i.e., when T is K 1,t for some t. We further show that the graph isomorphism problem is GI-complete and maximum clique APX-hard, when H contains a certain three-vertex multigraph as a minor. For the dominating set problem (parameterized by the size of H), we give FPT- and XPtime algorithms on star-graphs and H-graphs, respectively. 11

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13 1 Introduction: Geometric Representations of Graphs Geometric representations of graphs are an important tool to study various structural and algorithmic properties of graphs. A suitable representation of a graph can be used to understand the structure of a rather complicated graph. Since the very beginning of graph theory, the study of geometric representations is one of the main topics. In this thesis, we focus on intersection representations. An intersection representation R of a graph X is a collection {R v : v V (X)} such that uv E(X) if and only if R u R v ; the intersections encode the edges. To get nice graph classes, one typically restricts the sets R v to particular classes of geometrical objects. We give examples of important classes of intersection graphs in the following two sections. We also refer the reader to the classical books [38, 73]. In this chapter, we present several important definitions and an overview of our results. This thesis has two main parts: (i) the study of symmetries of geometrically represented graphs described by their automorphism groups, and (ii) algorithmic and complexity results for H-graphs. Part (i) is covered in Chapters 2 5. We show a general technique that allows one to determine the automorphism group of a graph if the structure of all intersection representations is well-understood. We apply this technique to interval graphs (Chapter 3), circle graphs (Chapter 4), and comparability graphs (Chapter 5). In particular, for interval graphs we get that their automorphism groups are exactly the same as the automorphism groups of trees. For other classes we give inductive characterizations of their automorphism groups using group products. In Section 1.1 we state those results precisely and describe the general approach. The whole part (ii) of this thesis is covered in Chapter 6. Here, we give the first comprehensive study of the complexity of important computational problems on H- graphs, introduced by Biró, Hujter, and Tuza [4]. Namely, we study the recognition, graph isomorphism, minimum dominating set, maximum clique, and coloring. Biró, Hujter, and Tuza asked whether it is possible to recognize H-graphs in polynomial time for a fixed graph H. We answer this question negatively. In particular, we prove that recognizing H-graphs is NP-complete when H is the complete graph on four vertices without an edge. For an introduction to this part see Section 1.2. A Comment on Notation. Groups and graphs are two main structures occurring in 13

14 Chapter 1. Introduction: Geometric Representations of Graphs this text. Usually, they are denoted by G in texts primarily concerning only one of them. In the first part of this thesis, both groups and graphs play an important role. We choose to denote groups by G, H and graphs by X, Y. The second part of the thesis is only concerned with graphs, so in this part we denote them by G, H. 1.1 Symmetries of Geometrically Represented Graphs The study of symmetries of geometrical objects is an ancient topic in mathematics and its precise formulation led to group theory. Symmetries play an important role in many distinct areas. In 1846, Galois used symmetries of the roots of a polynomial in order to characterize polynomials which are solvable by radicals. Automorphism Groups of Graphs. The symmetries of a graph X are described by its automorphism group Aut(X). Every automorphism is a permutation of the vertices which preserves adjacencies and non-adjacencies. Formally, a permutation π of V (X) is an automorphism if uv E(X) π(u)π(v) E(X). The automorphism group Aut(X) consists of all automorphisms of X. Frucht [29] proved that every finite group is isomorphic to Aut(X) of some graph X. General algebraic, combinatorial and topological structures can be encoded by (possibly infinite) graphs [43] while preserving automorphism groups. Most graphs are asymmetric, i.e., have only the trivial automorphism [24]. However, many mathematical results rely on highly symmetrical objects. Automorphism groups are important for studying large regular objects, since their symmetries allow one to simplify and understand these objects. Highly symmetrical large graphs with nice properties are often constructed algebraically from small graphs. Hoffman-Singleton graph is a 7-regular graph of diameter 2 with 50 vertices [44]. It has automorphisms and can be constructed from 25 copies of a small multigraph with 2 vertices and 7 edges [62, 71]. Similar constructions are used in designing large computer networks [25, 80]. For instance the well-studied degree-diameter problem asks, given integers d and k, to find a maximal graph X with diameter d and degree k. Such graphs are desirable networks having small degrees and short distances. Currently, the best constructions are highly symmetrical graphs made using groups [63]. In this thesis, we study automorphism groups of geometrically represented graphs. The main question we address is how the geometry influences their automorphism groups. For instance, the geometry of a sphere translates to 3-connected planar graphs which have unique embeddings [78]. Thus, their automorphism groups are so called spherical groups which are the automorphism groups of tilings of a sphere. For general planar graphs (PLANAR), the automorphism groups are more complex and they were described by Babai [1] and in more details in [51] by semidirect products of spherical and symmetric groups. Denition 1.1. For a graph class C, let Aut(C) = { G : X C, G = Aut(X) }. The class C is called universal if every abstract finite group is contained in Aut(C), and non-universal otherwise. 14

15 1.1. Symmetries of Geometrically Represented Graphs In 1869, Jordan [48] gave a characterization for the class of trees (TREE): Theorem 1.2 (Jordan [48]). The class Aut(TREE) is defined inductively as follows: (a) {1} Aut(TREE). (b) If G 1, G 2 Aut(TREE), then G 1 G 2 Aut(TREE). (c) If G Aut(TREE), then G S n Aut(TREE). The direct product in (b) constructs the automorphisms that act independently on non-isomorphic subtrees and the wreath product in (c) constructs the automorphisms that permute isomorphic subtrees. Graph Isomorphism Problem. This famous problem asks whether two input graphs X and Y are the same up to a relabeling. This problem is obviously in NP, and not known to be polynomially-solvable or NP-complete. Aside integer factorization, this is a prime candidate for an intermediate problem with the complexity between P and NPcomplete. It belongs to the low hierarchy of NP [70], which implies that it is unlikely NP-complete. (Unless the polynomial-time hierarchy collapses to its second level.) The graph isomorphism problem is known to be polynomially solvable for the classes of graphs with bounded degree [56] and with excluded topological subgraphs [40]. The graph isomorphism problem is the following fundamental mathematical question: given two mathematical structure, can we test their isomorphism in some more constructive way than by guessing a mapping and verifying that it is an isomorphism. The graph isomorphism problem is closely related to computing generators of an automorphism group. Assuming X and Y are connected, we can test X = Y by computing generators of Aut(X Y ) and checking whether there exists a generator which swaps X and Y. For the converse relation, Mathon [59] proved that generators of the automorphism group can be computed using O(n 3 ) instances of graph isomorphism. Compared to graph isomorphism, automorphism groups of restricted graph classes are much less understood. Regular Covering Testing. In [27], there was introduced a problem called regular graph covering testing which generalizes both graph isomorphism and Cayley graph testing. The input gives two graphs X and Y. The question is whether there exists a semiregular subgroup G of Aut(X) such that X/G = Y. It is shown in [27] that the problem can be solved in FPT-time with respect to the size of Y when X is planar. The general complexity of this problem is open, no NP-hardness reduction is known. To solve it, a good structural understanding of Aut(X) seems necessary. The characterizations of automorphisms groups in [52] and this paper can be used to attack this problem when X is an interval, circle or permutation graph Interval Graphs In an interval representation of a graph, each set R v is a closed interval of the real line. A graph is an interval graph if it has an interval representation; see Fig. 1.1a. A graph is a unit interval graph if it has an interval representation with each interval 15

16 Chapter 1. Introduction: Geometric Representations of Graphs (a) (b) Figure 1.1: (a) An interval graph and one of its interval representations. (b) A circle graph and one of its circle representations. of the length one. We denote these classes by INT and UNIT INT, respectively. Caterpillars (CATERPILLAR) are trees with all leaves attached to a central path; we have CATERPILLAR = INT TREE. Theorem 1.3. The following equalities hold: (i) Aut(INT) = Aut(TREE), (ii) Aut(connected UNIT INT) = Aut(CATERPILLAR), Concerning (i), this equality is not well known. It was stated by Hanlon [42] without a proof in the conclusion of his paper from 1982 on enumeration of interval graphs. Our structural analysis is based on PQ-trees [8] which describe all interval representations of an interval graph. It explains this equality and further solves an open problem of Hanlon: for a given interval graph, to construct a tree with the same automorphism group. Without PQ-trees, this equality is surprising since these classes are very different. Caterpillars which form their intersection have very restricted automorphism groups (see Lemma 3.6). The result (ii) follows from the known properties of unit interval graphs and our understanding of Aut(INT) Circle Graphs In a circle representation, each R v is a chord of a circle. A graph is a circle graph (CIRCLE) if it has a circle representation; see Fig. 1.1b. Theorem 1.4. Let Σ be the class of groups defined inductively as follows: (a) {1} Σ. (b) If G 1, G 2 Σ, then G 1 G 2 Σ. (c) If G Σ, then G S n Σ. (d) If G 1, G 2, G 3, G 4 Σ, then (G 4 1 G 2 2 G 2 3 G 2 4) Z 2 2 Σ. Then Aut(connected CIRCLE) consists of the following groups: If G Σ, then G Z n Aut(connected CIRCLE). If G 1, G 2 Σ, then (G n 1 G 2n 2 ) D n Aut(connected CIRCLE). 16

17 1.1. Symmetries of Geometrically Represented Graphs The automorphism group of a disconnected circle graph can be easily determined using Theorem 2.3. We are not aware of any previous results on the automorphism groups of circle graphs. We use split trees describing all representations of circle graphs. The class Σ consists of the stabilizers of vertices in connected circle graphs and Aut(TREE) Σ Comparability Graphs A comparability graph is derived from a poset by removing the orientation of the edges. Alternatively, every comparability graph X can be transitively oriented: if x y and y z, then xz E(X) and x z; see Fig 1.2a. This class was first studied by Gallai [32] and we denote it by COMP. An important structural parameter of a poset P is its Dushnik-Miller dimension [23]. It is the least number of linear orderings L 1,..., L k such that P = L 1 L k. (For a finite poset P, its dimension is always finite since P is the intersection of all its linear extensions.) Similarly, we define the dimension of a comparability graph X, denoted by dim(x), as the dimension of any transitive orientation of X. (Every transitive orientation has the same dimension; see Section 5.4.) By k-dim, we denote the subclass consisting of all comparability graphs X with dim(x) k. We get the following infinite hierarchy of graph classes: 1-DIM 2-DIM 3-DIM 4-DIM COMP. For instance, [69] proves that the bipartite graph of the incidence between the vertices and the edges of a planar graph always belongs to 3-DIM. Surprisingly, comparability graphs are related to intersection graphs, namely to function and permutation graphs. Function graphs (FUN) are intersection graphs of continuous real-valued function on the interval [0, 1]. Permutation graphs (PERM) are function graphs which can be represented by linear functions called segments [2]; see Fig. 1.2b and c. We have FUN = co-comp [39] and PERM = COMP co-comp = 2-DIM [26], where co-comp are the complements of comparability graphs. Since 1-DIM consists of all complete graphs, Aut(1-DIM) = {S n : n N}. The automorphism groups of 2-DIM = PERM are the following: Theorem 1.5. The class Aut(PERM) is described inductively as follows: (a) {1} Aut(PERM), (a) (b) (c) Figure 1.2: (a) A comparability graph with a transitive orientation. (b) A function graph and one of its representations. (c) A permutation graph and one of its representations. 17

18 Chapter 1. Introduction: Geometric Representations of Graphs (b) If G 1, G 2 Aut(PERM), then G 1 G 2 Aut(PERM). (c) If G Aut(PERM), then G S n Aut(PERM). (d) If G 1, G 2, G 3 Aut(PERM), then (G 4 1 G 2 2 G 2 3) Z 2 2 Aut(PERM). In comparison to Theorem 1.2, there is the additional operation (d) which shows that Aut(TREE) Aut(PERM). Geometrically, the group Z 2 2 in (d) corresponds to the horizontal and vertical reflections of a symmetric permutation representation. Notice that it is more restrictive than the operation (d) in Theorem 1.4. Colbourn [16] described an O(n 3 ) algorithm for graph isomorphism of permutation graphs. This was improved by Spinrad [72] to O(n 2 ) by computing modular decompositions and testing tree isomorphism on them. The bottleneck of this algorithm is computing the modular decomposition, so by combining with [61], the running time is improved to O(n + m). We are not aware of any previous polynomial-time algorithm for computing automorphism groups of permutation graphs. We describe a linear-time algorithm, by combining the modular decomposition and computing automorphism groups of trees. Corollary 1.6. There exists a linear-time algorithm for computing automorphism groups of permutation graphs. Further, using Theorem 1.5, our algorithm implicitly computes a structural decomposition of Aut(X), which is better and faster than outputting permutation generators of Aut(X). (These generators might be superlinear in size of X, but they can be computed compressed as in [17].) Also, our algorithm can be easily modified to solve graph isomorphism and canonization of permutation graphs, so it gives a more detailed description than [72]. Our result also easily gives the automorphism groups of bipartite permutation graphs (BIP PERM), in particular we have Aut(CATERPILLAR) Aut(BIP PERM) Aut(PERM). Corollary 1.7. The class Aut(connected BIP PERM) consists of all abstract groups G 1, G 1 Z 2 G 2 G 3, and (G 4 1 G 2 2) Z 2 2, where G 1 is a direct product of symmetric groups, and G 2 and G 3 are symmetric groups. Comparability graphs are universal since they contain bipartite graphs; we can orient all edges from one part to the other. Since the automorphism group is preserved by complementation, FUN = co-comp implies that also function graphs are universal. In Chapter 5, we explain the universality of FUN and COMP in more detail using the induced action on the set of all transitive orientations. Similarly posets are known to be universal [3, 76]. It is well-known that bipartite graphs have arbitrarily large dimensions: the crown graph, which is K n,n without a matching, has the dimension n. We give a construction which encodes any graph X into a comparability graph Y with dim(y ) 4, while preserving the automorphism group. 18

19 1.1. Symmetries of Geometrically Represented Graphs IFA CIRCLE CHOR co-4-dim FUN universal PLANAR PERM TRAPEZOID TREE BIP PERM INT CLAW-FREE non-universal CATERPILLAR UNIT INT co-bip Figure 1.3: The inclusions between the considered graph classes. We characterize the automorphism groups of the classes in gray. Theorem 1.8. For every k 4, the class k-dim is universal and its graph isomorphism is GI-complete. The same holds for posets of the dimension k. Yannakakis [79] proved that recognizing 3-DIM is NP-complete by a reduction from 3-coloring. For a graph X, a comparability graph Y is constructed with several vertices representing each element of V (X) E(X). It is proved that dim(y ) = 3 if and only if X is 3-colorable. Unfortunately, the automorphisms of X are lost in Y since it depends on the labels of V (X) and E(X) and Y contains some additional edges according to these labels. We describe a simple and completely different construction which achieves only the dimension 4, but preserves the automorphism group: for a given graph X, we create Y by replacing each edge with a path of length eight. However, it is non-trivial to show that Y 4-DIM, and the constructed four linear orderings are inspired by [79]. A different construction follows from [12, 77] Related Graph Classes Theorems 1.3, 1.4 and 1.5 and Corollary 1.7 state that INT, UNIT INT, CIRCLE, PERM, and BIP PERM are non-universal. We introduce several related classes of graphs. Figure 1.3 shows that many of those classes are already universal. Trapezoid graphs (TRAPEZOID) are intersection graphs of trapezoids between two parallel lines and they have universal automorphism groups [75]. Claw-free graphs (CLAW-FREE) are graphs with no induced K 1,3. Roberts [64] proved that UNIT INT = CLAW-FREE INT. The complements of bipartite graphs (co-bip) are claw-free and universal. Chordal graphs (CHOR) are intersection graphs of subtrees of trees. They contain no induced cycles of length four or more and naturally generalize interval graphs. Chordal graphs are universal [55]. Interval filament graphs (IFA) are intersection graphs of the following sets. For every R u, we choose an interval [a, b] and R u is a continuous function [a, b] R such that R u (a) = R u (b) = 0 and R u (x) > 0 for x (a, b). 19

20 Chapter 1. Introduction: Geometric Representations of Graphs Aut(M 1 ) Aut(M 2 ) 2 4 π = (3 4) 2 3 Figure 1.4: There are two different maps, depicted with the action of Aut(X). The stabilizers Aut(M i ) = Z 2 2 are normal subgroups. The remaining automorphisms morph one map into the other, for instance π transposing 2 and 3. We have Aut(X) = Z 2 2 Z Automorphism Groups Acting on Intersection Representations In this section, we describe the general technique which allows us to geometrically understand automorphism groups of some intersection-defined graph classes. Suppose that one wants to understand an abstract group G. Sometimes, it is possible interpret G using a natural action on some set which is easier to understand. The action is called faithful if no element of G belongs to all stabilizers. The structure of G is captured by a faithful action. We require that this action is faithful enough, which means that the stabilizers are simple and can be understood. Our approach is inspired by map theory. A map M is a 2-cell embedding of a graph; i.e, aside vertices and edges, it prescribes a rotational scheme for the edges incident with each vertex. One can consider the action of Aut(X) on the set of all maps of X: for π Aut(X), we get another map π(m) in which the edges in the rotational schemes are permuted by π; see Fig The stabilizer of a map M, called the automorphism group Aut(M), is the subgroup of Aut(X) which preserves/reflects the rotational schemes. Unlike Aut(X), we know that Aut(M) is always small (since Aut(M) acts semiregularly on flags) and can be efficiently determined. The action of Aut(X) describes morphisms between different maps and in general can be very complicated. Using this approach, the automorphism groups of planar graphs can be characterized [1, 51]. The Induced Action. For a graph X, we denote by Rep the set of all its (interval, circle, etc.) intersection representations. An automorphism π Aut(X) creates from R Rep another representation R such that R π(u) = R u; so π swaps the labels of the sets of R. We denote R as π(r), and Aut(X) acts on Rep. The general set Rep is too large. Therefore, we define a suitable equivalence relation and we work with Rep/. It is reasonable to assume that is a congruence with respect to the action of Aut(X): for every R R and π Aut(X), we have π(r) π(r ). We consider the induced action of Aut(X) on Rep/. The stabilizer of R Rep/, denoted by Aut(R), describes automorphisms inside this representation. For a nice class of intersection graphs, such as interval, circle or permutation graphs, the stabilizers Aut(R) are very simple. If it is a normal subgroup, then the quotient Aut(X)/Aut(R) describes all morphisms which change one representation in the orbit of R into another one. Our strategy is to understand these morphisms geometrically, for which we use the structure of all representations, encoded for the considered classes by PQ-, split and modular trees. 20

21 1.2. Algorithmic and Complexity Results for H-graphs 1.2 Algorithmic and Complexity Results for H-graphs Biró, Hujter, and Tuza [4] introduced the concept of an H-graph. Let H be a fixed graph. A graph G is an intersection graph of H if it is an intersection graph of connected subgraphs of H, i.e., for u, v V (G), the assigned subgraphs H v and H u of H share a vertex if and only if uv E(G). A subdivision H of a graph H is obtained when the edges of H are replaced by internally disjoint path of arbitrary lengths, i.e., an edge uv of H corresponds to a path from u to v in H such that all internal vertices on this path have degree two. A graph G is a topological intersection graph of H if G is an intersection graph of a subdivision H of H. We say that G is an H-graph and the collection {H v : v V (G)} of connected subgraphs of H is an H-representation of G. The class of all H-graphs is denoted by H-GRAPH. Additionally, a graph G is a Helly H-graph if it has an H-representation that satisfies the Helly property. These graph classes were introduced in the context of the (p, k) pre-coloring extension problem ((p, k)-prcolext). Here one is given a graph G together with a p-coloring of W V (G), and the goal is to find a proper k-coloring of G containing this pre-coloring. They showed that, for interval graphs, when k is part of the input (1, k)-prcolext can be solved in polynomial time, but (2, k)-prcolext is NP-complete. On the other hand, they provided an XP (in k and H ) algorithm to compute a (k, k)- PrColExt on H-GRAPH Related Graph Classes We discuss several known algorithmic results for some closely related classic graph classes. Interval Graphs. (See Section 1.1 for the definition.) The vast body of work involving interval graphs (and their generalizations) stems from the fact that many important computational problems can be solved efficiently on them. Recognition of interval graphs in linear time was a long-standing open problem solved by Booth and Lueker using PQ-trees [8] which can be used to describe the structure of all representations. Moreover, other important problems are solvable in linear time on interval graphs. These include, for example, the problem of finding a minimum dominating set [11], and the graph isomorphism problem [55]. Chordal Graphs. Recall that a graph is chordal when it does not have an induced cycle of length at least four. This definition is not particularly useful for algorithmic (a) (b) Figure 1.5: (a) A chordal graph and one of its representation as an intersection graph of subtrees of a tree. (b) A circular-arc graph and one of its representations. 21

22 Chapter 1. Introduction: Geometric Representations of Graphs results, but has given rise to many characterizations which are better suited to the task. For example one by Gavril [34], which states that a graph is chordal if and only if it can be represented as an intersection graph of subtrees of some tree; see Fig. 1.5a. An immediate consequence of this is that interval graphs form subclass of the chordal graphs. While the recognition problem can be solved easily in linear time for chordal graphs [66] and such algorithms can be used to generate an intersection representation by subtrees of a tree, asking for special host trees can be more difficult. For example, for a given graph G and a tree T it is NP-complete to decide whether G is a T - graph [49]. On the other hand, for a given graph G, if one would like to find a tree T with the fewest leaves such that G is a T -graph, it can be done in polynomial time [41] (this is known as the leafage problem). However, for any fixed d 3, if one would like to find a tree T where G is a T -graph and, for each vertex v, the subtree for v has at most d leaves, the problem again becomes NP-complete [13] (this is known as the d-vertex leafage problem). The minimum vertex leafage problem can be solved in n O(l) -time via a somewhat elaborate enumeration of minimal 1 tree representations of G with exactly l leaves where l is the leafage of G [13]. Additionally, some computational problems become harder on chordal graphs than interval graphs. Finding a minimum dominating set is NP-complete [7] on chordal graphs. The graph isomorphism problem is GI-complete on chordal graphs [55], i.e., it is as hard as the general graph isomorphism problem. On the other hand, the minimum coloring, maximum independent set, and maximum clique problems are all solvable in linear time on chordal graphs. An important subclass of chordal graphs are split graphs (SPLIT). A graph is a split graph if it can be partitioned into maximal clique and an independent set. Note that every split graph can be represented as an intersection graph of subtrees of a star S d, where S d is the complete bipartite graph K 1,d. Circular-Arc Graphs. These are a natural generalization of interval graphs. Here, each set R v corresponds to an arc of a circle; see Fig. 1.5b. This class is denoted by CIRCULAR-ARC. An important subclass of circular-arc graphs are Helly circulararc graphs. A graph G is a Helly circular-arc graph if the collection of circular arcs R = {R v : v V (G)} satisfies Helly property, i.e., for each sub-collection of R whose sets pairwise intersect, their common intersection is non-empty. Relations to H-GRAPH. Notice that we have the following relations: INT = K 2 -GRAPH, SPLIT S d -GRAPH, d=2 CIRCULAR-ARC = K 3 -GRAPH, and CHOR = T -GRAPH. Tree T 1 where each node of T corresponds to a maximal clique of G 22

23 1.2. Algorithmic and Complexity Results for H-graphs Recognition Graph isomorphism Dominating set Maximum clique Coloring S d -graphs T-graphs Helly H-graphs H-graphs O(n 4 ) n O( T 2 ) NP-completefor D-graphs GI-complete for some H FPTin d n O( T ) n O( H ) n O( H ) O(n+m) O(n+m) Polynomial APX-hard for some H, FPT O(n+m) O(n+m) FPT FPT Table 1.1: The table of the complexity of different problems for the four considered classes. Our results are highlighted. Hierarchy of H-GRAPH. Notice that, for any pair of (multi-)graphs H 1 and H 2, if H 1 is a minor of H 2, then H 1 -GRAPH H 2 -GRAPH. Additionally, if H 1 is a subdivision of H 2, then H 1 -GRAPH = H 2 -GRAPH. In particular, we have an infinite hierarchy of graph classes between interval and chordal graphs since INT CHOR, and for a tree T, we have T -GRAPH CHOR. Since some interesting computational problems are polynomial on interval graphs and hard on chordal graphs, an interesting question is the complexity of those problems on T -graphs, for a fixed tree T. Coloring H-GRAPH. Interestingly, it is NP-hard to compute a minimum coloring on Helly circular-arc graphs [35]. It follows that if H contains a cycle, then computing a minimum colouring in H-GRAPH is already NP-hard even for Helly H-GRAPH. Additionally, when H does not contain a cycle (i.e., H is a forest), H-GRAPH is a subclass of the chordal graphs, i.e., a minimum colouring can be computed in linear time. Therefore, for a graph H, it is NP-hard to compute the minimum chromatic number on graphs in the class (Helly) H-GRAPH when H contains a cycle, and solvable in linear time when H is acyclic Our Results Biró, Hujter, and Tuza ask the following question which we answer negatively. Problem 1.9 (Biró, Hujter, and Tuza [4], 1992). Let H be an arbitrary fixed graph. Is there a polynomial algorithm testing whether a given graph G is an H-graph? Moreover, we give a comprehensive study of H-graphs from the point of view of several other important problem in theoretical computer science such as graph isomorphism, dominating set, maximum clique, and coloring. We focus on four collections of classes of graphs: S d -GRAPH, T -GRAPH, Helly H-GRAPH, and H-GRAPH. Our results are displayed in Table 1.1. Recognition. We negatively answer the question of Biró, Hujter, and Tuza (Problem 1.9). In Theorem 6.13, we prove that recognizing D-graphs (D is the diamond graph) is NP-complete by a reduction from the problem of determining if the interval dimension of a partial order of height 1 is at most 3. For each tree T, we give a polynomial-time algorithm for recognizing T -graphs and O(n 4 )-time algorithm for recognizing S d -graphs (Theorem 6.5 and Theorem 6.7). 23

24 Chapter 1. Introduction: Geometric Representations of Graphs Graph Isomorphism. Theorem 6.24 shows that when H contains the three-vertex mutligraph where every two vertices are joined by two parallel edges as a minor, then graph isomorphism problem is GI-complete on H-graphs. Dominating Set. We solve the problem of finding a minimum dominating set on S d -graphs (Theorem 6.17) in O(d 2 d2 + n (n + m)) and for H-graphs (Theorem 6.18) in n O( H ). Maximum Clique. We show that the clique problem can be solved in polynomial time on Helly H-graphs (Theorem 6.23). Theorem 6.24 shows that for some H, the graph isomorphism problem is GI-complete and that maximum clique problem is APX-hard. Coloring. In Section 6.4, we use treewidth based methods to provide an FPTtime algorithm for k-coloring of H-graphs, and an FPT-time algorithm for finding a k-clique in an H-graph. 24

25 2 Preliminaries: Group Theory We describe some basic concepts of group theory that are essential for Chapters 3, 4, and 5. For a comprehensive treatment of the basics of group theory, see for example [67, 22], for a visual treatment of group theory, see [9]. We assume that the reader is familiar with the basic properties of groups. The following notation is used for the standard groups: S n is the symmetric group whose elements are n-element permutations, D n is the dihedral group whose elements are symmetries of the regular n-gon, including both rotations and reflections, Z n is the cyclic group whose elements are integers 0,..., n 1 and the operation is addition modulo n. 2.1 Group Products Group products allow decomposing of large groups into smaller ones. Consider for example the well known puzzle called the Rubik s Cube. The Rubik s Cube group G consists of all cube moves. It follows that G = 43, 252, 003, 274, 489, 856, The Rubik s Cube group is a huge object which seems to be very complicated. Using group products, one can understand the structure of this group: G = (Z 7 3 Z 11 2 ) ( (A 8 A 12 ) Z 2 ), where A n is the alternating group. One can combinatorially interpret the terms of the products and gain an insight into the structure of the Rubik s Cube. This can be used, for instance, to design algorithms solving it, or to understand the smallest number of moves necessary to solve it in any position. (Which is only 20.) Here, we explain two basic group theoretic methods for constructing larger groups from smaller ones, namely direct product and semidirect product. We show how these group operations can be used to construct automorphism groups of graphs. At the 25

26 Chapter 2. Preliminaries: Group Theory C 8 Aut(C8 ) = D 8 Figure 2.1: The cycle graph C 8 with the action of Aut(C 8 ) on its vertices and a Cayley graph of Aut(C 8 ). Note that Aut(C 8 ) is isomorphic to D 8. It is generated by two automorphisms: the rotation symmetry (depicted by the red arrows); the reflection symmetry (depicted by the blue arrows). end of this section, we prove Jordan s characterization of the automorphism groups of trees. Inspired by [9], we use Cayley graphs to visualize groups. Cayley graphs were actually invented by Cayley [10] for this purpose and now they also play an important role in combinatorial and geometric group theory. A Cayley graph is a colored oriented graph that depicts the abstract structure of a group. Suppose that G is a group and S is a generating set. The Cayley graph (G, S) is a graph constructed as follows: The elements of G correspond one-to-one to the vertices. Each generator s S is represented by a unique colour c(s). For every g G and s S, the there is a directed edge (g, gs) of colour c(s). Fig. 2.1 and 2.2 show examples of graphs and their automorphism groups represented by Cayley graphs Direct Product Let G and H be groups with operations G and H, respectively. Their direct product G H is a group having as elements all pairs (g, h) where g G and h H. The operation is defined componentwise: (g 1, h 1 ) (g 2, h 2 ) = (g 1 G g 2, h 1 H h 2 ). When there is no confusion we simply write (g 1 g 2, h 1 h 2 ) or (g 1 g 2, h 1 h 2 ). Fig. 2.3 shows an example. The direct product of n groups is defined similarly, and we use G n as a shorthand for the product G G G with n terms. Both G and H are normal subgroups of G H. On the other hand, the semidirect product, discussed in Section 2.1.2, constructs from two groups G and H a larger group for which only G is ensured to be a normal subgroup. The direct product can be used to construct automorphism groups of graphs that are disconnected and their connected components are pairwise non-isomorphic. In this 26

27 2.1. Group Products X Aut(X) = Z 8 Figure 2.2: A graph X with the automorphism group Aut(X) isomorphic to the group Z 8. This graph has only the rotation symmetries as automorphisms. Therefore, its automorphism group is isomorphic to a subgroup of Aut(C 8 ). We note that a similar gadget is used in the proof of Frucht s theorem to encode oriented colored edges of a Cayley graph. case, the automorphism group of a graph X is the direct product of the automorphism groups of its connected components X 1,..., X k : Aut(X) = Aut(X 1 ) Aut(X k ). The reason is that each automorphism acts independently on each component Semidirect and Wreath Products However, if we want to construct the automorphism group of a disconnected graph which has some isomorphic connected components, the direct product is not sufficient. The problem is that the automorphisms which permute the isomorphic components are not included in the direct product. We start with a simple example of two graphs in Fig The automorphism group of the graph X is isomorphic to S 3 Z 2, but the automorphism group of the graph Y is not Z 2 Z 2. The direct product does not include the automorphisms which swap the components. Moreover, Aut(Y ) is not even isomorphic to Z 2 Z 2 Z 2 because, for example, swapping the components and swapping the vertices of the left component do not commute. Semidirect Product. As already state, both G and H are normal subgroups of G H. The semidirect product generalizes the direct product since it only requires G to be a normal subgroup. This is one of the motivations for studying semidirect products since they allow to decompose a bigger number of groups. The direct product G H contains identical copies of G, with corresponding elements connected according to H, as shown in Fig In the semidirect product of Figure 2.3: A Cayley graph of the group Z 3 2. Note that the group contains two copies of Z 2 2, with the corresponding elements connected according to the group Z 2. 27

28 Chapter 2. Preliminaries: Group Theory X Figure 2.4: Two graphs X and Y. We have Aut(X) = S 3 Z 2, but we need the semidirect product to describe Aut(Y ). Y the groups G and H, the group H also determines how some copies of G are connected. However, these copies of G do not need to be all identical. First, we explain a special case: the semidirect product of the group G with its automorphism group Aut(G), denoted by G Aut(G). The elements are all pairs (g, f) such that g G and f Aut(G). The operation is defined in the following way: (g 1, f 1 ) (g 2, f 2 ) = (g 1 f 1 (g 2 ), f 1 f 2 ). Note that G Aut(G) defined like this forms a group. Its identity element is (1, 1) and the element (g, f) has the inverse (f 1 (g 1 ), f 1 ). We can think of it as all possible isomorphic copies of G connected according to Aut(G). The element (g 1, f 1 ) is in the isomorphic copy G 1 of G which we get by applying the automorphism f 1 on G. Multiplying (g 1, f 1 ) by (g 2, 1) corresponds to a movement inside G 1. Multiplying (g 1, f 1 ) by (1, f 2 ) corresponds to a movement from G 1 to the same elements of another isomorphic copy of G. In general, the semidirect product is defined for any two groups G and H, and a homomorphism ϕ: H Aut(G), denoted by G ϕ H. It is the set of all pairs (g, h) such that g G and h H. The operation is defined similarly to the operation of G Aut(G): (g 1, h 1 ) (g 2, h 2 ) = (g 1 ϕ(h 1 )(g 2 ), h 1 h 2 ). Again, it is quite straightforward to check that G ϕ H is a group. We can think of the homomorphism ϕ as if it assigns an isomorphic copy of G to each element of the group H. The isomorphic copies of G are then connected according to the group H. We write G H when there is no danger of confusion. Example 2.1. The dihedral group D 8 is equal to Z 8 Z 2. Fig. 2.1 on the right shows a Cayley graph of D 8. The elements of each of the two isomorphic copies of Z 8 are on one cycle, connected according to Z 2. Example 2.2. Let Y be the graph from Fig The group Aut(Y ) is isomorphic to Z 2 2 Z 2. Fig. 2.5 shows a Cayley graph of Aut(Y ). The elements of the two isomorphic copies of Z 2 2 are connected according to the pattern of Z 2. 28

29 2.1. Group Products Y Aut(Y) Figure 2.5: The structure of Aut(Y ), generated by three involutions acting on Y on the left: Aut(Y ) = Z 2 2 Z 2 = Z 2 Z 2. Wreath Product. The wreath product G H is a shorthand for the semidirect product G n ψ H where ψ is defined naturally by ψ(π) = (g 1,..., g n ) (g π(1),..., g π(n) ). In this thesis, we have H equal S n or Z n for which we use the natural actions on {1,..., n}. For more details, see [9, 67]. All semidirect products used in this paper are generalized wreath products of G 1,..., G k with H, in which each orbit of the action of H has assigned one group G i Automorphism Groups of Disconnected Graphs In 1869, Jordan described the automorphism groups of disconnected graphs, in terms of the automorphism groups of their connected components. Since a similar argument is used in several places in this paper, we describe his proof in details. Fig. 2.5 shows the automorphism group for a graph consisting of two isomorphic components. Theorem 2.3 (Jordan [48]). If X 1,..., X n are pairwise non-isomorphic connected graphs and X is the disjoint union of k i copies of X i, then Aut(X) = Aut(X 1 ) S k1 Aut(X n ) S kn. Proof. Since the action of Aut(X) is independent on non-isomorphic components, it is clearly the direct product of factors, each corresponding to the automorphism group of one isomorphism class of components. It remains to show that if X consists of k isomorphic components of a connected graph Y, then Aut(X) = Aut(Y ) S k. We isomorphically label the vertices of each component. Then each automorphism π Aut(X) is a composition σ τ of two automorphisms: σ maps each component to itself, and τ permutes the components as in π while preserving the labeling. Therefore, the automorphisms σ can be bijectively identified with the elements of Aut(Y ) k and the automorphisms τ with the elements of S k. Let π, π Aut(X). Consider the composition σ τ σ τ, we want to swap τ with σ and rewrite this as a composition σ ˆσ ˆτ τ. Clearly the components are permuted in π π exactly as in τ τ, so ˆτ = τ. On the other hand, ˆσ is not necessarily equal σ. Let σ be identified with the vector (σ 1,..., σ k ) Aut(Y )k. Since σ is applied after τ, 29