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

TL;DR

This paper investigates the properties of $ riangle$-convex sets in graphs, focusing on the Carathéodory and exchange numbers across different graph families and products.

Contribution

It introduces the concepts of Carathéodory and exchange numbers in the context of $ riangle$-convexity and analyzes these parameters for various graph classes and products.

Findings

Determined Carathéodory numbers for specific graph families.

Analyzed exchange numbers in different graph products.

Provided bounds and exact values for these parameters.

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.