Split Edge Geodetic Domination Number of a Graph

Abstract In this paper, we introduce a new graph theoretic parameter, split edge geodetic domination number of a connected graph as follows. A set S ⊆ V(G) is said to be a split edge geodetic dominating set of G if S is both a split edge geodetic set and a dominating set of G ( < V-S > is disconnected). The minimum cardinality of the split edge geodetic dominating set of G is called split edge geodetic domination number of G and is denoted by γ 1g s (G). It is shown that for any 3 positive integers m, f and nwith 2 ≤ m ≤ f ≤ n-2, there exists a connected graph G of order n such that g 1 (G) = m and γ 1g s (G) = f. For every pair l, n of integers with 2 ≤ l ≤ n-2, there exists a connected graph G of order n such that γ 1g s (G) = l.