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.