On the geodetic hull number of P-free graphs

Abstract

We show the hardness of the geodetic hull number for chordal graphs.

The Geodetic Hull Number is Hard for Chordal Graphs

“…Using a polynomial reduction from [LO1] [8], it has been shown in [5] that Satisfiability restricted to such instances is still NP-complete.…”

“…One of the most well studied convexity notions for graphs is the shortest path convexity or geodetic convexity, where a set X of vertices of a graph G is considered convex if no vertex outside of S lies on a shortest path between two vertices inside of S. Defining the convex hull of a set S of vertices as the smallest convex set containing S, a natural parameter of G is its hull number h(G) [7], which is the minimum order of a set of vertices whose convex hull is the entire vertex set of G. The hull number is NP-hard for bipartite graphs [2], partial cubes [1], and P 9 -free graphs [5], but it can be computed in polynomial time for cographs [4], (q, q − 4)-graphs [2], {paw, P 5 }-free graphs [3,5], and distance-hereditary graphs [9]. Bounds on the hull number are given in [2,6,7].…”

“…Let G be a graph, and let S be a set of vertices of G. The interval I G (S) of S in G is the set of all vertices of G that lie on shortest paths in G between vertices from S. Note that S ⊆ I G (S), and that if and only if S ∩ I G ({v, w}) = ∅ for every two vertices v and w in V (G) \ S. The hull H G (S) of S in G, defined as the smallest convex set in G that contains S, equals the intersection of all convex sets that contain S. The set S is a hull set if H G (S) = V (G), and the hull number h(G) of G [5,7] is the smallest order of a hull set of G.…”

A polynomial time algorithm for geodetic hull number for complementary prisms

“…Given a graph G and an integer k, the problem of deciding whether h(G) ≤ k is NP-complete for a general graph G, even if G is bipartite [3], partial cube [2], chordal [5], or P 9 -free [16]. On the other hand, such a parameter can be determined in polynomial time for unit interval graphs, cographs, split graphs [15], (q, q − 4)graphs [3], {paw, P 5 }-free graphs [4,16], and distance-hereditary graphs [24]. Coelho et al [12] provide additional references concerning the hull number, also in other graph convexities.…”

On the geodetic hull number for complementary prisms II

“…[3,10,18]. The computation of hull number is NP-hard for bipartite graphs [3], partial cubes [1], and P 9 -free graphs [12], but it can be computed in polynomial time for cographs, split graphs [9], (q, q−4)-graphs [3], {paw, P 5 }-free graphs [12], and distance-hereditary graphs [19].…”