Minimal path convexity under some graph operations

Abstract

Let G be a connected graph. A path P in G is called an m-path if the graph induced by the vertex set V(P) of P is P. A subset C of V(G) is said to be m-convex if, for every pair of vertices x, y ε C, the vertex set of every x-y m-path is contained in C. The cardinality of a maximal proper m-convex set in G is the m-convexity number of G. This paper characterizes the m-convex sets of graphs under some graph operations, and gives the convexity number of these graphs.