Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, th...

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. It is known that if T is a tree of order n, then γr(T ) ≥ d(n+2)/3e. In this note we provide a simple constructive characteriz...

We investigate the domination number and total domination number of the graph K q (n; k) whose vertices are all the k-subspaces of an n-dimensional vector space over a eld with q elements and whose edges are the pairs fU; Wg of vertices such that U \ W = f0g. Bounds are obtained in general and exact results are obtained for n k 2 +k?1 and in other cases when q is suuciently large relative to n ...

Using hypergraph transversals it is proved that γt(Qn+1) = 2γ(Qn), where γt(G) and γ(G) denote the total domination number and the domination number of G, respectively, and Qn is the n-dimensional hypercube. More generally, it is shown that if G is a bipartite graph, then γt(G K2) = 2γ(G). Further, we show that the bipartiteness condition is essential by constructing, for any k > 1, a (non-bipa...