The Hull Number in the Convexity of Induced Paths of Order 3

Abstract

A set S of vertices of a graph G is $P^*_3$-convex if there is no vertex outside S having two non-adjacent neighbors in S. The $P^*_3$-convex hull of S is the minimum $P^*_3$-convex set containing S. If the $P^*_3$-convex hull of S is V(G), then S is a $P^*_3$-hull set. The minimum size of a $P^*_3$-hull set is the $P^*_3$-hull number of G. In this paper, we show that the problem of deciding whether the $P^*_3$-hull number of a chordal graph is at most k is NP-complete and present a linear-time algorithm to determine this parameter and provide a minimum $P^*_3$-hull set for unit interval graphs.