Let G = (V,E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the ...

The domination polynomial of a graph G of order n is the polynomial D(G, x) = Pn i=γ(G) d(G, i)x , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper, we obtain some properties of the coefficients of D(G, x). Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by G′(m), we obtain a rela...

A directed dominating set in a directed graph D is a set S of vertices of V such that every vertex u ∈ V (D) \ S has an adjacent vertex v in S with v directed to u. The directed domination number of D, denoted by γ(D), is the minimum cardinality of a directed dominating set in D. The directed domination number of a graph G, denoted Γd(G), which is the maximum directed domination number γ(D) ove...

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A set D of vertices of a graph G is a dominating set if every vertex in V \D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam h...

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...

In this paper we introduce the concept of strong total domination in fuzzy graphs. We determine the strong total domination number for several classes of fuzzy graphs. A lower bound and an upper bound for the strong total domination number in terms of strong domination number is obtained. Strong total domination in fuzzy trees is studied. A necessary and sufficient condition for the set of fuzz...

We find the maximum number of edges for a graph of given order and value of parameter for several domination parameters. In particular, we consider the total domination and independent domination numbers.

The paired bondage number (total restrained bondage number, independent bondage number, k-rainbow bondage number) of a graph G, is the minimum number of edges whose removal from G results in a graph with larger paired domination number (respectively, total restrained domination number, independent domination number, k-rainbow domination number). In this paper we show that the decision problems ...

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we disprove a conjecture posed in a recent paper(On an open problem concerning total domination critical graphs, Expo. Mat...