The closed geodetic numbers of the corona and composition of graphs
Abstract
Given two vertices u and v of a connected graph G, the closed interval IG[u,v;] is that set of all vertices lying in some u - v geodesic in G. If S ⊆ V(G), then IG[S] = ∪{IG[u,v] :u,v ∈ S}. A set S of vertices in G is called a geodetic cover of G if I G[S] = V(G). Suppose that in constructing a geodetic cover S of G, we select a vertex v1 and let S1 = {v1}. Select a vertex v2 ≠ v1 and let S2 = {v1, v2}. Then successively select a vertex vi ∉ I G[Si-1] and let Si = {v1, v 2,..., vi} until there is some positive integer k for which Sk = S. Any such geodetic cover of G obtained from this way is called the closed geodetic cover of G. The closed geodetic number and the upper closed geodetic number of G are, respectively, the smallest and the largest cardinality among closed geodetic covers of G. A closed geodetic cover S of G is a minimal closed geodetic cover of G if no proper subset of 5 is a closed geodetic cover of G. The minimal closed geodetic number is the maximum cardinality of a minimal closed geodetic cover of G. It is shown that every three positive integers m,k and n with 2 ≤ m < k < n are realizable as the closed geodetic, minimal closed geodetic and upper closed geodetic numbers, respectively, of a connected graph. This present paper determines the closed geodetic, upper closed geodetic and minimal closed geodetic numbers of graphs resulting from a corona and a composition of connected graphs.
