Total Outer Independent Geodetic Domination Number of a Graph

In this paper the concept of the total outer independent geodetic domination number of a graph G is introduced. An outer independent geodetic dominating set S  V(G) is said to be a total outer independent geodetic dominating set of G if the subgraph has no isolated vertices. The minimum cardinality of a total outer independent geodetic dominating set is called the total outer independent geodetic domination number and is denoted by gtoi(G). Some general properties satisfied by this concept are studied. The total outer independent geodetic domination number of certain classes of graphs are determined. It is shown that for every pair m, n of integers with 3 ≤ m ≤ n, there exist a connected graph G of order n such that gtoi(G) = m. Also, it is shown that for any three integers p, q and r such that 2 ≤ p ≤ q  r there exists a connected graph G with g(G) = p , g(G) = q and gtoi (G) = r.