Several of the best known problems and conjectures in graph theory arise in studying the behavior of a graphical invariant on a graph product. Examples of this are Vizing’s conjecture, Hedetniemi’s conjecture and the calculation of the Shannon capacity of graphs, where the invariants are the domination number, the chromatic number and the independence number on the Cartesian, categorical and st...

For a connected graph G with at least two vertices and S a subset of vertices, the convex hull [S]G is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V (G) with [S]G = V (G). Upper bound for the hull number of strong product G⊠H of two graphs G and H is obtainted. Improved upper bounds are obtained for some class of strong product gr...

The strong isometric dimension idim(G) of a graph G is the least number k such that G can be isometrically embedded into the strong product of k paths. The problem of determining idim(G) for graphs of diameter two is reduced to a covering problem of the complement of G with complete bipartite graphs. As an example it is shown that idim(P ) = 5, where P is the Petersen graph.

We give various reformulations of the Strong Perfect Graph Conjecture, based on a study of forced coloring procedures, uniquely colorable subgraphs and ! ? 1-cliques in minimal imperfect graphs.

We will extend Reed's Semi-Strong Perfect Graph Theorem by proving that unbreakable C 5-free graphs diierent from a C 6 and its complement have unique P 4-structure.

The Strong Perfect Graph Conjecture, suggested by Claude Berge in 1960, had a major impact on the development of graph theory over the last forty years. It has led to the definitions and study of many new classes of graphs for which the Strong Perfect Graph Conjecture has been verified. Powerful concepts and methods have been developed to prove the Strong Perfect Graph Conjecture for these spec...

In this paper we give a characterization of kernel-perfect (and of critical kernel-imperfect) arc-local tournament digraphs. As a consequence, we prove that arc-local tournament digraphs satisfy a strenghtened form of the following interesting conjecture which constitutes a bridge between kernels and perfectness in digraphs, stated by C. Berge and P. Duchet in 1982: a graph G is perfect if and ...

In this article, we present a characterization of basic graphs in terms of forbidden induced subgraphs. This class of graphs was introduced by Conforti, Cornuéjols and Vušković [3], and it plays an essential role in the announced proof of the Strong Perfect Graph Conjecture by Chudnovsky, Robertson, Seymour and Thomas [2]. Then we apply the Reducing Pseudopath Method [13] to characterize the su...

in this paper, we investigate a problem of finding natural condition to assure the product of two graphs to be hamilton-connected. we present some sufficient and necessary conditions for $gbox h$ being hamilton-connected when $g$ is a hamilton-connected graph and $h$ is a tree or $g$ is a hamiltonian graph and $h$ is $k_2$.

The Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.