On the Geodetic and Hull Numbers of Some Graphs

Abstract

Let G be a connected graph, u and v be vertices of G and I[u, v] the closed interval consisting of u, v and all vertices lying on some u-v geodesic. If S ⊆ V(G), then I[S] is the union of all sets I[u, v] for all u, v ∈ S. A subset S of V(G) is called a geodetic set in G if I[S] = V(G). The minimum cardinality of a geodetic set in G is called the geodetic number of G. The convex hull [S] of a subset S of  V(G) is defined as the smallest convex set in G containing S. The minimum cardinality among the subsets S of V(G) with [S] = V(G) is called the hull number of G. In this paper, we give the geodetic number and the hull number of some graphs.