In this paper we focus on 2- domination number of a fuzzy graph G by using effective edge and is denoted γ2(G) obtain some results concept, the relationship between other concepts are obtained.

Let f be a function that assigns to each vertex a subset of colors chosen from a set C = {1, 2, . . . , k} of k colors. If  u∈N(v) f (u) = C for each vertex v ∈ V with f (v) = ∅, then f is called a k-rainbow dominating function (kRDF) of G where N(v) = {u ∈ V | uv ∈ E}. The weight of f , denoted by w(f ), is defined as w(f ) =  v∈V |f (v)|. Given a graph G, the minimum weight among all weight...

4 In this paper, we study the domination number, the global dom5 ination number, the cographic domination number, the global co6 graphic domination number and the independent domination number 7 of all the graph products which are non-complete extended p-sums 8 (NEPS) of two graphs. 9

A dominating set in a graph G is a set S of vertices such that every vertex outside S has a neighbor in S; the domination number γ(G) is the minimum size of such a set. The independent domination number, written i(G), is the minimum size of a dominating set that also induces no edges. Henning and Southey conjectured that if G is a connected cubic graph with sufficiently many vertices, then i(G)...

We let γ(G) and i(G) denote the domination number and the independent domination number ofG, respectively. Recently, Rad and Volkmann conjectured that i(G)/γ(G) ≤ ∆(G)/2 for every graph G, where ∆(G) is the maximum degree of G. In this note, we construct counterexamples of the conjecture for ∆(G) ≥ 6, and give a sharp upper bound of the ratio i(G)/γ(G) by using the maximum degree of G.

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...

It is known that the problem of computing adjacency dimension a graph NP-hard. This suggests finding for special classes graphs or obtaining good bounds on this invariant. In work we obtain general G in terms parameters . We discuss tightness these and, some particular graphs, closed formulae. particular, show close relationships exist between and other parameters, like domination number, locat...