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

On Box-Perfect Graphs

Let $G=(V,E)$ be a graph and let $A_G$ be the clique-vertex incidence matrix of $G$. It is well known that $G$ is perfect iff the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is totally dual integral (TDI). In 1982, Cameron and Edmonds proposed to call $G$ box-perfect if the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is box-totally dual integral (box-TDI), and posed the problem of characterizing such graphs. In this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity graphs, and identify several other classes of box-perfect graphs. We also develop a general and powerful method for establishing box-perfectness

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

Counting Weighted Independent Sets beyond the Permanent

Jerrum, Sinclair, and Vigoda [J. ACM, 51 (2004), pp. 671--697] showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. Equivalently, this may be viewed as approximating the partition function of vertex-weighted independent sets in the line graph of a bipartite graph. Line graphs of bipartite graphs are perfect graphs and are known to be precisely the class of (claw, diamond, odd hole)-free graphs. So how far does the result of Jerrum, Sinclair, and Vigoda extend? We first show that it extends to (claw, odd hole)-free graphs, and then show that it extends to the even larger class of (fork, odd hole)-free graphs. Our techniques are based on graph decompositions, which have been the focus of much recent work in structural graph theory, and on structural results of Chvátal and Sbihi [J. Combin. Theory Ser. B, 44 (1988)], Maffray and Reed [J. Combin. Theory Ser. B, 75 (1999)], and Lozin and Milanič [J. Discrete Algorithms, 6 (2008), pp. 595--604]

Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results

Forbidden Substructures in Graphs and Trigraphs, and Related Coloring Problems

Given a graph G, χ(G) denotes the chromatic number of G, and ω(G) denotes the clique number of G (i.e. the maximum number of pairwise adjacent vertices in G). A graph G is perfect provided that for every induced subgraph H of G, χ(H) = ω(H). This thesis addresses several problems from the theory of perfect graphs and generalizations of perfect graphs. The bull is a five-vertex graph consisting of a triangle and two vertex-disjoint pendant edges; a graph is said to be bull-free provided that no induced subgraph of it is a bull. The first result of this thesis is a structure theorem for bull-free perfect graphs. This is joint work with Chudnovsky, and it first appeared in [12]. The second result of this thesis is a decomposition theorem for bull-free perfect graphs, which we then use to give a polynomial time combinatorial coloring algorithm for bull-free perfect graphs. We remark that de Figueiredo and Maffray [33] previously solved this same problem, however, the algorithm presented in this thesis is faster than the algorithm from [33]. We note that a decomposition theorem that is very similar (but slightly weaker) than the one from this thesis was originally proven in [52], however, the proof in this thesis is significantly different from the one in [52]. The algorithm from this thesis is very similar to the one from [52]. A class G of graphs is said to be χ-bounded provided that there exists a function f such that for all G in G, and all induced subgraphs H of G, we have that χ(H) ≤ f(ω(H)). χ-bounded classes were introduced by Gyarfas [41] as a generalization of the class of perfect graphs (clearly, the class of perfect graphs is χ-bounded by the identity function). Given a graph H, we denote by Forb*(H) the class of all graphs that do not contain any subdivision of H as an induced subgraph. In [57], Scott proved that Forb*(T) is χ-bounded for every tree T, and he conjectured that Forb*(H) is χ-bounded for every graph H. Recently, a group of authors constructed a counterexample to Scott's conjecture [51]. This raises the following question: for which graphs H is Scott's conjecture true? In this thesis, we present the proof of Scott's conjecture for the cases when H is the paw (i.e. a four-vertex graph consisting of a triangle and a pendant edge), the bull, and a necklace (i.e. a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge). This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [13]. Finally, we consider several operations (namely, "substitution," "gluing along a clique," and "gluing along a bounded number of vertices"), and we show that the closure of a χ-bounded class under any one of them, as well as under certain combinations of these three operations (in particular, the combination of substitution and gluing along a clique, as well as the combination of gluing along a clique and gluing along a bounded number of vertices) is again χ-bounded. This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [14]