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

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