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

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*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.