The coloring problem is studied in the paper for graph classes defined by two small forbidden induced subgraphs. We prove some sufficient conditions for effective solvability of the problem in such classes. As their corollary we determine the computational complexity for all sets of two connected forbidden induced subgraphs with at most five vertices except 13 explicitly enumerated cases.

We characterize the class of intersection graphs of hypergraphs with a rank of at most 3 and a multiplicity of at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. 2000 Mathematics Subject Classification: 05C62, 05C75, 05C70, 05C65

Let P be an induced hereditary property and L(P) denote the class of all graphs that satisfy the property P locally. The purpose of the present paper is to describe the minimal forbidden subgraphs of L(P) and the structure of local properties. Moreover, we prove that L(P) is irreducible for any hereditary property P. c © 2001 Elsevier Science B.V. All rights reserved.

Let IG(x, y) and JG(x, y) be the geodesic and induced path interval between x and y in a connected graph G, respectively. The following three betweenness axioms are considered for a set V and R : V × V → 2 (i) x ∈ R(u, y), y ∈ R(x, v), x 6= y, |R(u, v)| > 2 ⇒ x ∈ R(u, v) (ii) x ∈ R(u, v) ⇒ R(u, x) ∩R(x, v) = {x} (iii)x ∈ R(u, y), y ∈ R(x, v), x 6= y,⇒ x ∈ R(u, v). We characterize the class of g...

A class of graphs is Kruskalian if Kruskal’s theorem on a well-quasi-ordering of finite trees provides a finite characterization in terms of forbidden induced subgraphs. Let k be a natural number. A graph is a k-cograph if its vertices can be colored with colors from the set {1, . . . ,k} such that for every nontrivial subset of vertices W there exists a partition {W1,W2} of W into nonempty sub...

A magnet is a pair u, v of adjacent vertices such that the proper neighbours of u are completely linked to the proper neighbours of v. It has been shown that one can reduce the graph by removing one of the vertices of a magnet together with all edges linking the other one to its proper neighbours without changing the stability number. We characterize a class of graphs such that by repeated use ...

A graph G is coordinated if, for every induced subgraph H of G, the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex. In a previous work, coordinated graphs were characterized by minimal forbidden induced subgraphs within some ...

The linear-width of a graph G is de ned to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e1; : : : ; er) in such a way that for every i = 1; : : : ; r 1, there are at most k vertices incident to edges that belong both to fe1; : : : ; eig and to fei+1; : : : ; erg. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal for...

For a hereditary property P let kP(G) denote the number of forbidden subgraphs contained in G. A graph G is said to be weakly Psaturated, if G has the property P and there is a sequence of edges of G, say e1, e2, . . . , el, such that the chain of graphs G = G0 ⊂ G0+e1 ⊂ G1 + e2 ⊂ . . . ⊂ Gl−1 + el = Gl = Kn (Gi+1 = Gi + ei+1) has the following property: kP(Gi+1) > kP(Gi), 0 ≤ i ≤ l − 1. In thi...