A Note on Convex Basic Graphs

Abstract

A graph G is convex basic if the convex subsets of the vertex set V ( G) of G are all trivial. In this paper, shall give some characterizations of convex basic graphs. Specifically, we shall relate convex basic graphs with the concepts such as hull set and convexity number of a graph. Convex basic graphs resulting from the sum, composition, and cartesian product of graphs are also characterized. As one of our results, we show that for any positive integer p, the set of all connected graphs with independence number p contains only a finite number of convex basic graphs and an infinite number of non-convex
basic graphs.