In a graph G, a vertex subset S ⊆ V (G) is said to be a dominating set of G if every vertex not in S is adjacent to a vertex in S. A dominating set S of a graph G is called a paired-dominating set if the induced subgraph G[S] contains a perfect matching. The paired-domination problem involves finding a smallest paired-dominating set of G. Given an intersection model of an interval graph G with ...

An odd dominating set of a simple, undirected graph G = (V,E) is a set of vertices D ⊆ V such that |N [v]∩D| ≡ 1 mod 2 for all vertices v ∈ V . It is known that every graph has an odd dominating set. In this paper we consider the concept of connected odd dominating sets. We prove that the problem of deciding if a graph has a connected odd dominating set is NP-complete. We also determine the exi...

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 double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = (V,E), a subset D ⊆ V (G) is a 2dominating set if every vertex of V (...

We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth. Graphs of bounded growth are used to characterize wireless communication networks, and this class of graph includes many models known from the literature, e.g. (Quasi) Unit Disk Graphs. An independent dominating set is a dominating set in ...

Many graph problems cannot be computed in polynomial time unless P = NP , which most computer scientists and mathematicians doubt. Examples are the Traveling Salesman Problem (TSP), vertex cover, and dominating set. TSP is to find a Hamiltonian cycle with the least weight in a complete weighted graph. A vertex cover of a graph is a subset of the vertices that covers all edges. A dominating set ...

Let be a simple undirected fuzzy graph. A subset S of V is called a dominating set in G if every vertex in V-S is effectively adjacent to at least one vertex in S. A dominating set S of V is said to be a Independent dominating set if no two vertex in S is adjacent. The independent domination number of a fuzzy graph is denoted by (G) which is the smallest cardinality of a independent dominating ...

Let G = (V, E) be a simple graph. A dominating set D is called a complementary tree dominating set if the induced subgraph is a tree. The minimum cardinality of a complementary tree dominating set is called the complementary tree domination number of G and is denoted by ctd(G). For a graph G, let V(G) = {v : v  V(G)} be a copy of V(G). The splitting graph Sp(G) of G is the graph with ...

1 I n t r o d u c t i o n A subset D C V is a dominating set in graph G = (V, E) iff for all vertices v E V \ D there is a vertex u E D with uv E E. It is a dominating clique in G iff D is a dominating set in G and a clique (i.e. for all u, v E D uv E E). There are many papers investigating the problem of finding minimum dominating sets in graphs with (and without) additional requirements to th...

In this paper, we consider r-dominating cliques in homogeneously orderable graphs (a common generalization of dually chordal and distance-hereditary graphs) and their relation to strict r-packing sets. We prove that a homogeneously orderable graph G possesses an r-dominating clique if and only if for any pair of vertices x , y of G d (x , y ) ° r ( x ) / r ( y ) / 1 holds where r : V r N is a g...

A Mobile Ad Hoc Network (MANET) is an infrastructureless wireless network that can support highly dynamic mobile units. The multi-hop feature of a MANET suggests the use of clustering to simplify routing. Graph domination can be used in defining clusters in MANETs. A variant of dominating set which is more suitable for clustering MANETs is the weakly-connected dominating set. A cluster is defin...