On hereditary graph classes defined by forbidding Truemper configurations: recognition and combinatorial optimization algorithms, and χ-boundedness results

Truemper configurations are four types of graphs that helped us understand the structure of several well-known hereditary graph classes. The most famous examples are perhaps the class of perfect graphs and the class of even-hole-free graphs: for both of them, some Truemper configurations are excluded (as induced subgraphs), and this fact appeared to be useful, and played some role in the proof of the known decomposition theorems for these classes. The main goal of this thesis is to contribute to the systematic exploration of hereditary graph classes defined by forbidding Truemper configurations. We study many of these classes, and we investigate their structure by applying the decomposition method. We then use our structural results to analyze the complexity of the maximum clique, maximum stable set and optimal coloring problems restricted to these classes. Finally, we provide polynomial-time recognition algorithms for all of these classes, and we obtain χ-boundedness results

Algorithms for square-3PC(·, ·)-free Berge graphs

We consider the class of graphs containing no odd hole, no odd antihole, and no configuration consisting of three paths between two nodes such that any two of the paths induce a hole, and at least two of the paths are of length 2. This class generalizes clawfree Berge graphs and square-free Berge graphs. We give a combinatorial algorithm of complexity O(n7) to find a clique of maximum weight in such a graph. We also consider several subgraph-detection problems related to this class

The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Algorithms for square-3PC(.,.)-free Berge graphs

We consider the class of graphs containing no odd hole, no odd antihole and no configuration consisting of three paths between two nodes such that any two of the paths induce a hole and at least two of the paths are of length 2. This class generalizes claw-free Berge graphs and square-free Berge graphs. We give a combinatorial algorithm of complexity O(n7) to find a clique of maximum weight in such a graph. We also consider several subgraph-detection problems related to this class.Recognition algorithm, maximum weight clique algorithm, combinatorial algorithms, perfect graphs, star decompositions.

Structure and algorithms for (cap, even hole)-free graphs

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most [View the MathML source], and hence [View the MathML source], where ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set, where n is the number of vertices and m is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G)−1 and clique-width at most 48

Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family

Structural Characterisations of Hereditary Graph Classes and Algorithmic Consequences

A hole is a chordless cycle of length at least four, and is even or odd depending onthe parity of its length. Many interesting classes of graphs are defined by excluding (possibly among other graphs) holes of certain lengths. Most famously perhaps is the class of Berge graphs, which are the graphs that contain no odd hole and no complement of an odd hole. A graph is perfect if the chromatic number of each of its induced subgraphs is equal to the size of a maximum clique in that subgraph. It was conjectured in the 1960’s by Claude Berge that Berge graphs and perfect graphs are equivalent, that is, a graph is perfect if and only if it is Berge. This conjecture was finally resolved by Chudnovsky, Robertson, Seymour and Thomas in 2002, and it is now called the strong perfect graph theorem. Graphs that do not contain even holes are structurally similar to Berge graphs, and for this reason Conforti, Cornuéjols, Kapoor and Vušković initiated the study of even-hole-free graphs. One of their main results was a decomposition theorem and a recognition algorithm for even-hole-free graphs, and many techniques developed in the pursuit of a decomposition theorem for even-hole-free graphs proved useful in the study of perfect graphs. Indeed, the proof of the strong perfect graph theorem relied on decomposition, and many interesting graph classes have since then been understood from the viewpoint of decomposition. In this thesis we study several classes of graphs that relate to even-hole-free graphs. First, we focus on β-perfect graphs, which form a subclass of even-hole-free graphs. While it is unknown whether even-hole-free graphs can be coloured in polynomial time, β-perfect graphs can be coloured optimally in polynomial time using the greedy colouring algorithm. The class of β-perfect graphs was introduced in 1996 by Markossian, Gasparian and Reed, and since then several classes of β-perfect graphs have been identified but no forbidden induced subgraph characterisation is known. In this thesis we identify a new class of β-perfect graphs, and we present forbidden induced subgraph characterisations for the class of β-perfect hyperholes and for the class of claw-free β-perfect graphs. We use these characterisations to decide in polynomial time whether a given hyperhole, or more generally a claw-free graph, is β-perfect. A graph is l-holed (for an integer l ≥ 4) if every one of its holes is of length l. Another focus of the thesis is the class of l-holed graphs. When l is odd, the l-holed graphs form a subclass of even-hole-free graphs. Together with Preissmann, Robin, Sintiari, Trotignon and Vušković we obtained a structure theorem for l-holed graphs where l ≥ 7. Working independently, Cook and Seymour obtained a structure theorem for the same class of graphs. In this thesis we establish that these two structure theorems are equivalent. Furthermore, we present two recognition algorithms for l-holed graphs for odd l ≥ 7. The firs uses the structure theorem of Preissmann, Robin, Sintiari, Trotignon, Vušković and the present author, and relies on decomposition by a new variant of a 2-join called a special 2-join, and the second uses the structure theorem of Cook and Seymour, and relies only on a process of clique cutset decomposition. We also give algorithms that solve in polynomial time the maximum clique and maximum stable set problems for l-holed graphs for odd l ≥ 7. Finally, we focus on circular-arc graphs. It is a long standing open problem to characterise in terms of forbidden induced subgraphs the class of circular-arc graphs, and even the class of chordal circular-arc graphs. Motivated by a result of Cameron, Chap-lick and Hoàng stating that even-hole-free graphs that are pan-free can be decomposed by clique cutsets into circular-arc graphs, we investigate the class of even-hole-free circular-arc graphs. We present a partial characterisation for the class of even-hole-free circular-arc graphs that are not chordal