The Structure of Claw-Free Perfect Graphs

In 1988, Chvátal and Sbihi [4] proved a decomposition theorem for claw-free perfect graphs. They showed that claw-free perfect graphs either have a clique-cutset or come from two basic classes of graphs called elementary and peculiar graphs. In 1999, Maffray and Reed [6] successfully described how elementary graphs can be built using line-graphs of bipartite graphs using local augmentation. However gluing two claw-free perfect graphs on a clique does not necessarily produce claw-free graphs. In this paper we give a complete structural description of claw-free perfect graphs. We also give a construction for all perfect circular interval graphs

The Structure of Claw-Free Perfect Graphs

Abstract In 1988, Chvátal and Sbihi [4] proved a decomposition theorem for claw-free perfect graphs. They showed that claw-free perfect graphs either have a clique-cutset or come from two basic classes of graphs called elementary and peculiar graphs. In 1999, Maffray and Reed [6] successfully described how elementary graphs can be built using line-graphs of bipartite graphs using local augmentation. However gluing two claw-free perfect graphs on a clique does not necessarily produce claw-free graphs. In this paper we give a complete structural description of claw-free perfect graphs. We also give a construction for all perfect circular interval graphs

Balancedness of subclasses of circular-arc graphs

A graph is balanced if its clique-vertex incidence matrix contains no square submatrix of odd order with exactly two ones per row and per column. There is a characterization of balanced graphs by forbidden induced subgraphs, but no characterization by mininal forbidden induced subgraphs is known, not even for the case of circular-arc graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. In this work, we characterize when a given graph G is balanced in terms of minimal forbidden induced subgraphs, by restricting the analysis to the case where G belongs to certain classes of circular-arc graphs, including Helly circular-arc graphs, claw-free circular-arc graphs, and gem-free circular-arc graphs. In the case of gem-free circular-arc graphs, analogous characterizations are derived for two superclasses of balanced graphs: clique-perfect graphs and coordinated graphs.Fil: Bonomo, Flavia. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Duran, Guillermo Alfredo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Universidad de Chile; ChileFil: Safe, Martin Dario. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Wagler, Annegret Katrin. Centre National de la Recherche Scientifique; Franci

Graph Structure and Coloring

We denote by G=(V,E) a graph with vertex set V and edge set E. A graph G is claw-free if no vertex of G has three pairwise nonadjacent neighbours. Claw-free graphs are a natural generalization of line graphs. This thesis answers several questions about claw-free graphs and line graphs. In 1988, Chvatal and Sbihi proved a decomposition theorem for claw-free perfect graphs. They showed that claw-free perfect graphs either have a clique-cutset or come from two basic classes of graphs called elementary and peculiar graphs. In 1999, Maffray and Reed successfully described how elementary graphs can be built using line graphs of bipartite graphs and local augmentation. However gluing two claw-free perfect graphs on a clique does not necessarily produce claw-free graphs. The first result of this thesis is a complete structural description of claw-free perfect graphs. We also give a construction for all perfect circular interval graphs. This is joint work with Chudnovsky. Erdos and Lovasz conjectured in 1968 that for every graph G and all integers s,t≥ 2 such that s+t-1=χ(G) > ω(G), there exists a partition (S,T) of the vertex set of G such that ω(G|S)≥ s and χ(G|T)≥ t. This conjecture is known in the graph theory community as the Erdos-Lovasz Tihany Conjecture. For general graphs, the only settled cases of the conjecture are when s and t are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stability number 2, line graphs and quasi-line graphs. The second part of this thesis considers the conjecture for claw-free graphs and presents some progresses on it. This is joint work with Chudnovsky and Fradkin. Reed's ω, ∆, χ conjecture proposes that every graph satisfies χ≤ ⎡½ (Δ+1+ω)⎤ ; it is known to hold for all claw-free graphs. The third part of this thesis considers a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colorings that achieve our bounds: The complexity are O(n²) for line graphs and O(n³m²) for quasi-line graphs. For line graphs, this is faster than the best known algorithm for constructing a coloring that achieves the bound of Reed's original conjecture. This is joint work with Chudnovsky, King and Seymour

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

Perfect Matchings in Claw-free Cubic Graphs

Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for claw-free graphs.Comment: 6 pages, 2 figure