On Cliques and Forcing $m$-Convexity Numbers of Graphs

Abstract

This paper introduces the concepts of forcing $m$-convexity number and forcing clique number of a graph. We show that the forcing $m$-convexity numbers of some Cartesian product and composition of graphs are related to the forcing clique numbers of the graphs. We also show that the forcing $m$-convexity number of the composition $G[K_n]$, where $G$ is a connected graph with no extreme vertex, is equal to the forcing $m$-convexity number of $G$.