The hull domination number of a graph

Abstract

For a connected graph G = (V, E), the hull number h(G) of a graph G is the minimum cardinality of a set of vertices whose convex hull contains all vertices of G. A hull set S in a connected graph G is called a minimal hull set of G if no proper subset of S is a hull set of G.A subset D of vertices in G is called a dominating set if every vertex not in D has at least one neighbor in D. A Hull dominating set M is both a Hull and a dominating set. The hull (domination, hull domination) number h(G),(γ(G),γh(G)) of G is the minimum cardinality among all hull (dominating, hull dominating) sets in G. Hull domination number of certain classes of graphs are determined. Connected graph of order p with hull domination number p or p-1 are characterized. It is shown that for every two integers a,b≥2 with 2≤a≤b, there is a connected graph G such that γh (G)=a and γg (G)=b, where γg (G) is the geodetic domination number of a graph.