A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. In earlier papers of this series we proved that every claw-free graph either belongs to one of several basic classes that we described explicitly, or admits one of a few kinds of decomposition. In this paper we convert this “decomposition” theorem into a theorem describing the global structure of claw-free graphs.

In this paper, we present several conditions for K1;3-free graphs, which guarantee the graph is subpancyclic. In particular, we show that every K1;3-free graph with a minimum degree sum 2 ¿ 2 √ 3n+ 1 − 4; every {K1;3; P7}-free graph with 2 ¿ 9; every {K1;3; Z4}-free graph with 2 ¿ 9; and every K1;3-free graph with maximum degree , diam(G)¡ ( +6)=4 and 2 ¿ 9 is subpancyclic. c © 2002 Elsevier Sc...

We give a strengthening of the closure concept for claw-free graphs introduced by the second author in 1997. The new closure of a claw-free graph G de ned here is uniquely determined and preserves the value of the circumference of G. We present an in nite family of graphs with n vertices and 3 2 n 1 edges for which the new closure is the

This paper studies slide-resisting forces generated by claws in the free-walking beetle Pachnoda marginata (Coleoptera, Scarabaeoidea) with emphasis on the relationship between the dimension of the claw tip and the substrate texture. To evaluate the force range by which the claw can interact with a substrate, forces generated by the freely moving legs were measured using a load cell force trans...

Some known results on claw-free (K 1;3-free) graphs are generalized to the larger class of almost claw-free graphs which were introduced by Ryjj a cek. In particular, we show that a 2-connected almost claw-free graph is 1-tough, and that a 2-connected almost claw-free graph on n vertices is hamiltonian if 1 3 (n ? 2), thereby (partly) generalizing results of Matthews and Sumner. Finally, we use...

Matchings in graphs correspond to independent sets in the corresponding line graphs. Line graphs are an important subclass of claw-free graphs. Hence studying independence polynomials of claw-free graphs is a natural extension of studying matching polynomials of graphs. We extend a result of Bayati et.al. showing a fully polynomial time approximation scheme (FPTAS) for computing the independenc...

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. How...

A graph is prismatic if for every triangle T , every vertex not in T has exactly one neighbour in T . In a previous paper we gave a complete description of all 3-colourable prismatic graphs, and of a slightly more general class, the “orientable” prismatic graphs. In this paper we describe the non-orientable ones, thereby completing a description of all prismatic graphs. Since complements of pri...

A graph G is Nm-locally connected if for every vertex v in G, the vertices not equal to v and with distance at most m to v induce a connected subgraph in G. We show that both connectedN2-locally connected claw-free graph and 3-edge-connected N3-locally connected claw-free graph have connected even [2, 4]-factors, which settle a conjecture by Li in [6].