Carath$\'{e}$odory Number and Exchange Number in $\Delta$-convexity

Abstract

Given a graph $G$, a set is $\Delta$-convex if there is no vertex $u\in V(G)\setminus S$ forming a triangle with two vertices of $S$. The $\Delta$-convex hull of $S$ is the minimum $\Delta$-convex set containing $S$. This article is an attempt to discuss the Carath\'eodory number and exchange number on various graph families and standard graph products namely Cartesian, strong and, lexicographic products of graphs.