A necessary condition for the equality of the clique number and the convexity number of a graph

Abstract

Let G be a graph, u and v two vertices of G and X a subset of V(G). A u−vgeodesic is a path between u and v of minimum length. Ig(u,v) is the set of vertices that lie on any u−v geodesic and Ig(X) is the set ⋃u,v∈XIg(u,v). X is g-convex if Ig(X)=X. The convexity number, con(G), of G is the maximum cardinality of a proper g-convex set of G. The clique number, ω(G), of G is the maximum cardinality of a clique of G. If G is a connected not complete graph then ω(G)≤con(G). In this paper a necessary condition for ω(G)=con(G) is provided and, on the basis of this condition, both the class of distance-hereditary graphs for which ω(G)=con(G) and the class of chordal graphs for which ω(G)=con(G) are characterized.