On the Carathéodory Number for the Convexity of Paths of Order Three

Abstract

Abstract Let G be a finite, simple, and undirected graph and let S be a set of vertices of G. If no vertex of G that does not belong to S has two neighbours in S, then S is P 3 -convex. The P 3 -convex hull H G ( S ) of S is the smallest P 3 -convex set containing S. The P 3 -Caratheodory number of G is the smallest integer c such that for every set S and every vertex u in H G ( S ) , there is a set F ⊆ S with | F | ⩽ c and u ∈ H G ( F ) . We study structural and algorithmic aspects of the P 3 -Caratheodory number. We characterize the P 3 -Caratheodory number of trees and block graphs, establish upper bounds on the P 3 -Caratheodory number of general graphs and of claw-free graphs and prove that it is NP-complete to decide for a given bipartite graph G and a given integer k, whether the P 3 -Caratheodory number of G is at least k.