A subset X of edges in a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X. The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. Let m,n and k be positive integers with n − 1 ≤ m ≤ (n 2 ) , G(m,n) be the set of all non-isomorphic connected graphs of order n and size m, and G(m,n; k) = {G ∈ G(m...

In this paper, we evaluate the connected domination number of (Zn), in some case of n. We find out that the connected domination number of (Z p e1 1 ×p e2 2 ×···×p ek k ) is equal to k. Finally, we characterize the graphs in which γ ( (Zn)) = γc( (Zn)). AMS subject classification: 05C25, 05C69.

A subset S of V is called a total dominating set if every vertex in V is adjacent to some vertex in S. The total domination number γt (G) of G is the minimum cardinality taken over all total dominating sets of G. A dominating set is called a connected dominating set if the induced subgraph 〈S〉 is connected. The connected domination number γc(G) of G is the minimum cardinality taken over all min...

For a given graph G = (V,E), a set D ⊆ V (G) is said to be an outerconnected dominating set if D is dominating and the graph G−D is connected. The outer-connected domination number of a graph G, denoted by γ̃c(G), is the cardinality of a minimum outer-connected dominating set of G. We study several properties of outer-connected dominating sets and give some bounds on the outer-connected dominati...

A connected domination set of a graph is a set of vertices such that every vertex not in is adjacent to and the induced subgraph of is connected. The minimum connected domination set of a graph is the connected domination set with the minimum number of vertices. In this paper, we propose parallel algorithms for finding the minimum connected domination set of interval graphs and circular-arc gra...

An outer-connected dominating set for an arbitrary graph G is a set D̃ ⊆ V such that D̃ is a dominating set and the induced subgraph G[V \ D̃] be connected. In this paper, we focus on the outerconnected domination number of the product of graphs. We investigate the existence of outer-connected dominating set in lexicographic product and Corona of two arbitrary graphs, and we present upper bounds f...