Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [9,14] and claw-free graphs [8,9]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [10]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far,...

Graphs with circular symmetry, called webs, are relevant for describing the stable set polytopes of two larger graph classes, quasi-line graphs [6,10] and claw-free graphs [5,6]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a longstanding problem [7]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it i...

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [8,12] and claw-free graphs [7,8]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] v...

We show that deciding whether a given graph G of size m has a unique perfect matching as well as finding that matching, if it exists, can be done in time O(m) if G is either a cograph, or a split graph, or an interval graph, or claw-free. Furthermore, we provide a constructive characterization of the claw-free graphs with a unique perfect matching.

The second author’s ω, ∆, χ conjecture proposes that every graph satisties χ ≤ d 1 2 (∆ + 1 + ω)e. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way we discuss a stronger local conjecture, and prove that it holds for claw-free graphs with a three-colourable complement. To prove our results ...

If G is a claw-free graph of sufficiently large order n, satisfying a degree condition σk > n+k2−4k+7 (where k is an arbitrary constant), then G has a 2-factor with at most k − 1 components. As a second main result, we present classes of graphs C1, . . . , C8 such that every sufficiently large connected claw-free graph satisfying degree condition σ6(k) > n + 19 (or, as a corollary, δ(G) > n+19 ...

We propose an algorithm for solving the maximum weighted stable set problem on claw-free graphsthat runs in O(n)−time, drastically improving the previous best known complexity bound. This algorithm is based on a novel decomposition theorem for claw-free graphs, which is also introduced in the present paper. Despite being weaker than the well-known structure result for claw-free graphs given...

In [3], the closure cl(G) for a claw-free graph G is de ned, and it is proved that G is hamiltonian if and only if cl(G) is hamiltonian. On the other hand, there exist in nitely many claw-free graphs G such that G is not hamiltonian-connected (resp. homogeneously traceable) while cl(G) is hamiltonian-connected (resp. homogeneously traceable). In this paper we de ne a new closure clk(G) (k 1) as...

Matthews and Sumner have proved in [10] that if G is a 2-connected claw-free graph of order n such that δ(G) ≥ (n − 2)/3, then G is Hamiltonian. We say that a graph is almost claw-free if for every vertex v of G, 〈N(v)〉 is 2-dominated and the set A of centers of claws of G is an independent set. Broersma et al. [5] have proved that if G is a 2-connected almost claw-free graph of order n such th...

Let L be a set of graphs. Free(L) is the set of graphs that do not contain any graph in L as an induced subgraph. It is known that if L is a set of four-vertex graphs, then the complexity of the coloring problem for Free(L) is known with three exceptions: L = {claw, 4K1}, L = {claw, 4K1, co-diamond}, and L = {C4, 4K1}. In this paper, we study the coloring problem for Free(claw, 4K1). We solve t...