A total dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D. A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D. The total (double, respectively) domination number of a graph G is the min...

Let G be an IFG. Then V D  is said to bae a strong (weak) dominating set if every D V v   is strongly (weakly) dominated by some vertex in D. We denote the strong (weak) intuitionistic fuzzy dominating set by sid-set (wid-set). The minimum vertex cardinality over all the sid-set (wid-set) is called the strong (weak) dominating number of an IFG and is denoted by )] ( [ ) ( G G wid sid   In ...

Given a graph G = (V,E), a perfect dominating set is a subset of vertices V ′ ⊆ V (G) such that each vertex v ∈ V (G) \ V ′ is dominated by exactly one vertex v ∈ V . An efficient dominating set is a perfect dominating set V ′ where V ′ is also an independent set. These problems are usually posed in terms of edges instead of vertices. Both problems, either for the vertex or edge variant, remain...

Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E′ ⊆ E which dominates all edges of G. In particular, if every edge of G is dominated by exactly one edge of E′ then E′ is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to de...

Let p be a positive integer and G = (V,E) a simple graph. A p-dominating set of G is a subset S of V such that every vertex not in S is dominated by at least p vertices in S. The p-domination number γp(G) is the minimum cardinality among the p-dominating sets of G. In this paper, for p ≥ 2, we give three equivalent conditions for trees with unique minimum p-dominating sets and also give a const...

Efficient routing among a set of mobile hosts (also called nodes) is one of the most important functions in ad hoc wireless networks. Routing based on a connected dominating set is a promising approach, where the search space for a route is reduced to the nodes in the set. A set is dominating if all the nodes in the system are either in the set or neighbors of nodes in the set. In this paper, w...

Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E′ ⊆ E which dominates all edges of G. In particular, if every edge of G is dominated by exactly one edge of E′ then E′ is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to de...

We conduct an extensive numerical analysis of the effects of network damage on dominating sets that occur at either randomly selected nodes or at the highest-degree nodes, in synthetic scale-free and real complex networks. We focus on the fraction of nodes that remain dominated after the damage has occurred, defined as stability. We find that minimum dominating sets (MDS), despite being optimal...