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THE UPPER RESTRAINED GEODETIC DOMINATION NUMBER OF A GRAPH

A set S of vertices of a connected graph G is a restrained geodetic dominating set, if either S = V or S is a geodetic dominating set with the subgraph G[V − S] induced by V − S has no isolated vertices. The minimum cardinality of a restrained geodetic dominating set of G is called the restrained geodetic domination number and is denoted by γgr(G). A restrained geodetic dominating set S in a connected graphG is called a minimal restrained geodetic dominating set of G if no proper subset of S is a restrained geodetic dominating set of G. The upper restrained geodetic domination number γ gr(G) is the maximum cardinality of a minimal restrained geodetic dominating set of G. The upper restrained geodetic domination number of certain classes of graphs are determined. It is shown that for every pair of integers a, b with 3 ≤ a ≤ b, there exists a connected graph G of order b such that γ gr(G) = a. Also, for any four integers a, b, c and d with 2 ≤ a ≤ b ≤ c ≤ d ≤ p, there exists a connected graph G of order p such that g(G) = a, γg(G) = b, γgr(G) = c and γ gr(G) = d.