Let tt : (V, E, W ) be a finite, connected, weighted graph without loops and multiple edges. In a weighted graph each arc is assigned a weight by the weight function W : E +. A u v path P in tt is called a weighted u v geodesic if the weighted distance between u and v is calculated along P . The strength of a path is the minimum weight of its arcs, and length of a path is the number of edges in...

We show the hardness of the geodetic hull number for chordal graphs.

In this paper we learn the new idea of forcing total outer independent edge geodetic number a graph. Let G be connected graph and R minimum set G. A subset L ⊆ is known as for if unique containing L. cardinality R. The denoted by (G) = min , where taken over all in Some general properties satisfied concept are studied. It shown that any couple integers l, m with0 < l ≤ − 4, there exists such .

Given a connected graph and a vertex x ∈ V (G), the geodesic graph Px(G) has the same vertex set as G with edges uv iff either v is on an x − u geodesic path or u is on an x − v geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of Km,n with itself, where m,n ≥ 4, is e...

Let G be a connected graph and S ⊆ V (G). Then the Steiner distance of S, denoted by dG(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I (S) is the union of all vertices that belong to some Steiner tree for S. If S = {u, v}, then ...