The hull number in the convexity of induced paths of order 3

Abstract

A set S of vertices of a graph G is P3⁎-convex if there is no vertex outside S having two non-adjacent neighbors in S. The P3⁎-convex hull of S is the minimum P3⁎-convex set containing S. If the P3⁎-convex hull of S is V(G), then S is a P3⁎-hull set. The minimum size of a P3⁎-hull set is the P3⁎-hull number of G. In this paper, we show that the problem of deciding whether the P3⁎-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 P3⁎-hull set for unit interval graphs.