A vertex subset S of a graph G is a dominating set if every vertex of G either belongs to S or is adjacent to a vertex of S. The cardinality of a smallest dominating set is called the dominating number of G and is denoted by γ(G). A graph G is said to be γ-vertex-critical if γ(G− v) < γ(G), for every vertex v in G. Let G be a 2-connected K1,5-free 3-vertex-critical graph. For any vertex v ∈ V (...

A mixed dominating set S of a graph G = (V,E) is a subset S ⊆ V ∪ E such that each element v ∈ (V ∪ E) \ S is adjacent or incident to at least one element in S. The mixed domination number γm(G) of a graph G is the minimum cardinality among all mixed dominating sets in G. The problem of finding γm(G) is know to be NP-complete. In this paper, we present an explicit polynomial-time algorithm to c...

A vertex subset S of a graph G is a dominating set if every vertex of G either belongs to S or is adjacent to a vertex of S. The cardinality of a smallest dominating set is called the dominating number of G and is denoted by γ(G). A graph G is said to be γ-vertex-critical if γ(G− v) < γ(G), for every vertex v in G. Let G be a 2-connected K1,5-free 3-vertex-critical graph. For any vertex v ∈ V (...

A cycle C in a graph G is said to be dominating if E(G−C) = ∅. Enomoto et al. showed that if G is a 2-connected triangle-free graph with α(G) ≤ 2κ(G) − 2, then every longest cycle is dominating. But it is unknown whether the condition on the independence number is sharp. In this paper, we show that if G is a 2-connected triangle-free graph with α(G) ≤ 2κ(G) − 1, then G has a longest cycle which...

We consider an efficient distributed algorithm for determining a dominating and absorbant set of vertices (mobile hosts) in a given directed graph that represents an ad hoc wireless network with unidirectional links. This approach is based on the concept of dominating set in graph theory. A host v is called a dominating neighbor (absorbant neighbor) of another host u if there is a directed edge...

A graph $G$ is called $P_4$-free, if $G$ does not contain an induced subgraph $P_4$. The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x)=sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,x)$ is called a domination root of $G$. In this paper we state and prove formula for the domination polynomial of no...

Let G = (V, E) be a graph. We define strong dominating color number of a graph G as the maximum number of color classes which are strong dominating sets of G, and is denoted by sd(G) , where the maximum is taken over all  -coloring of G. This paper determines the exact values of strong dominating color number of sum and Cartesian product of graphs.

Preprocessing by data reduction is a simple but powerful technique used for practically solving different network problems. A number of empirical studies shows that a set of reduction rules for solving Dominating Set problems introduced by Alber, Fellows & Niedermeier leads efficiently to optimal solutions for many realistic networks. Despite of the encouraging experiments, the only class of gr...

For any graph, there is a largest integer k such that given any partition of the vertex set with at most k elements in each class of the partition, there is transversal of the partition that is a dominating set in the graph. Some basic results about this parameter, the partition domination number, are obtained. In particular, it is shown that its value is 2 for the two-dimensional infinite grid...

A set D of vertices in 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) is the minimum cardinality of a dominating set of G. A dominating set D of a graph G is total dominating set if the induced subgraph 〈D〉 has no isolated vertices. In this paper, we introduce the total co-independent domination in graphs, exact value for some s...