In this paper, we study both concepts of geodetic dominatingand edge geodetic dominating sets and derive some tight upper bounds onthe edge geodetic and the edge geodetic domination numbers. We also obtainattainable upper bounds on the maximum number of elements in a partitionof a vertex set of a connected graph into geodetic sets, edge geodetic sets,geodetic domin...

a subset $s$ of vertices in a graph $g$ is called a geodetic set if every vertex not in $s$ lies on a shortest path between two vertices from $s$‎. ‎a subset $d$ of vertices in $g$ is called dominating set if every vertex not in $d$ has at least one neighbor in $d$‎. ‎a geodetic dominating set $s$ is both a geodetic and a dominating set‎. ‎the geodetic (domination‎, ‎geodetic domination) number...

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)...

Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),(1≤i≠j≤n). The maximum cardinality of π...

A subset S of vertices in a graph G is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A subset D of vertices in G is called dominating set if every vertex not in D has at least one neighbor in D. A geodetic dominating set S is both a geodetic and a dominating set. The geodetic (domination, geodetic domination) number g(G)(γ(G), γg(G)) of G is...

A edge 2-rainbow dominating function (E2RDF) of a graph G is a ‎function f from the edge set E(G) to the set of all subsets‎ ‎of the set {1,2} such that for any edge.......................

An edge geodetic set of a connected graph G of order p ≥ 2 is a set S ⊆ V (G) such that every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum cardinality of its edge geodetic sets and any edge geodetic set of cardinality g1(G) is a minimum edge geodetic set of G or an edge geodetic basis of G. An edge geodetic set S in ...

The edge C4 graph of a graph G, E4(G) is a graph whose vertices are the edges of G and two vertices in E4(G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C4. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C4 graphs are isomorphic. We study the relationship between the diameter, radius and domina...