In this paper, we introduce a new graph theoretic parameter, split edge geodetic domination number of connected as follows. A set S ⊆ V(G) is said to be dominating G if both and ( < V-S > disconnected). The minimum cardinality the called denoted by γ1gs(G). It shown that for any 3 positive integers m, f nwith 2 ≤ m n-2, there exists order n such g1 (G) = γ1gs f. For every pair l, with l γ1gs(G)...

We show that the geodetic number of proper interval graphs can be computed in polynomial time. This problem is NP-hard on chordal graphs and on bipartite weakly chordal graphs. Only an upper bound on the geodetic number of proper interval graphs has been known prior to

Buckley and Harary introduced several graphical invariants related to convexity theory, such as the geodetic number of a graph. These invariants have been the subject of much study and their determination has been shown to be NP -hard. We use the probabilistic method developed by Erdös to determine the asymptotic behavior of the geodetic number of random graphs with fixed edge probability. As a...

For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W -tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steine...

A set S ⊆ V (G) is called a geodetic if every vertex of G lies on shortest u-v path for some u, v ∈ S, the minimum cardinality among all sets number and denoted by . C chromatic contains vertices different colors in G, geo-chromatic Sc both set. The G. In this paper, we determine 2-cartesian product standard graphs like complete graphs, cycles paths.