A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γpr(G), is the minimum cardinality of a paired-dominating set of G. In [1], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, ...

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γpr(G), is the minimum cardinality of a paired-dominating set of G. In [?], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, ...

We review a characterization of domination perfect graphs in terms of forbidden induced subgraphs obtained by Zverovich and Zverovich [12] using a computer code. Then we apply it to a problem of unique domination in graphs recently proposed by Fischermann and Volkmann. 1 Domination perfect graphs Let G be a graph. A set D ⊆ V (G) is a dominating set of G if each vertex of G either belongs to D ...

We present a polynomial-time approximation scheme (PTAS) for the minimum dominating set problem in unit disk graphs. In contrast to previously known approximation schemes for the minimum dominating set problem on unit disk graphs, our approach does not assume a geometric representation of the vertices (specifying the positions of the disks in the plane) to be given as part of the input. The alg...

A subset D of the vertex set V (G) of a graph G is called a dominating set of G if every vertex in V − D is adjacent to a vertex in D. The minimum cardinality of a dominating set is called the domination number and is denoted by γ(G). A dominating set D such that δ(< D >) = 0 is called an isolate dominating set. The minimum cardinality of an isolate dominating set is called the isolate dominati...

DOMINATING SET remains NP -complete even when instances are restricted to bipartite graphs, however, in this case VERTEX COVER is solvable in polynomial time. Consequences to VECTOR DOMINATING SET as a generalization of both are discussed.

In this paper, we discuss the approximability of the capacitated b-edge dominating set problem, which generalizes the edge dominating set problem by introducing capacities and demands on the edges. We present an approximation algorithm for this problem and show that it achieves a factor of 8/3 for general graphs and a factor of 2 for bipartite graphs. Moreover, we discuss the relationships of t...

Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset E of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of E. We say that E is a perfect edge dominating set of G, if every edge not in E is dominated by exactly one edge of E. The perfect edge dominating problem is to determine a least cardinality perf...

A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159–162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D ∪ T necessarily contains all vertices of the graph.