On the Complexity of the P3-Hull Number of the Cartesian Product of Graphs

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 neighbors in S, then S is P3-convex. The P3-convex hull H(S) of S is the smallest P3-convex set containing S. If H(S)=V(G) we say that S is a P3-hull set of G. The cardinality h(G) of a minimum P3-hull set in G is called the P3-hull number of G. In this paper w extend the result of Centeno et al. [Theoretical Computer Science 412 (2011), 3693-3700] showing that, given a graph G and an integer k, deciding whether h(G)≤k remains NP-complete for the Cartesian product of graphs.