Many hard graph problems, such as Hamiltonian Cycle, become FPT when parameterized by treewidth, a parameter that is bounded only on sparse graphs. When parameterized by the more general parameter cliquewidth, Hamiltonian Cycle becomes W[1]-hard, as shown by Fomin et al. [5]. Sæther and Telle address this problem in their paper [13] by introducing a new parameter, split-matching-width, which li...

The class of DCT-graphs is a common generalization of the classes of almost claw-free and quasi claw-free graphs. We prove that every even (2p + 1)-connected DCT-graph G is p-extendable, i.e., every set of p independent edges of G is contained in a perfect matching of G. This result is obtained as a corollary of a stronger result concerning factor-criticality of DCT-graphs.

In this paper we summarize known results on claw-free graphs. The paper is subdivided into the following chapters and sections: 1. Introduction 2. Paths, cycles, hamiltonicity a) Preliminaries b) Degree and neighborhood conditions c) Local connectivity conditions d) Further forbidden subgraphs e) Invariants f) Squares g) Regular graphs h) Other hamiltonicity related results and generalizations ...

We say that a graph G is k-extendable if every set of k independent edges of G can be extended to a perfect matching. In the paper it is proved that if G is an even (2k + 1)-connected K 1;k+3-free graph such that the set of all centers of claws is independent, then G is k-extendable. As a corollary we obtain an analogous result for almost claw-free graphs and for claw-free graphs, thus extendin...

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γpr(G), is the minimum cardinality of a paired-dominating set of G. In [1], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, ...

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γpr(G), is the minimum cardinality of a paired-dominating set of G. In [?], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, ...

We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose elements give invariants of alternating links which strictly refine the first fo...

A matching-cut of a graph is an edge cut that matching. The problem recognizing graphs with and NP-complete, even if the belongs to one number classes. We initiate study for without fixed path as induced subgraph. show in P P5-free graphs, but there exists integer t>0 which it NP-complete Pt-free graphs.