the broadcast domination number of the cartesian product of two cycles is completely determined.

A paired-dominating set of a graph is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph. Recently, Chen, Sun and Xing [Acta Mathematica Scientia Series A Chinese Edition 27(1) (2007), 166–170] proved that a cubic graph has paired-domination number at most three-fifths ...

Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V (G) is a p-dominating set of the graph G, if every vertex v ∈ V (G)−D is adjacent with at least p vertices of D. The p-domination number γp(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ1(G) is the usual domination number γ(G). If G is a nontrivial connected block graph,...

A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class), where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F -coloring of a graph is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of...

For a graph $G$ let $gamma (G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets...

A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. A dominating set for G is a connected dominating set if it induces a connected subgraph of G. The connected domination number of G, denoted by γc(G), is the minimum cardinality of a connected dominating set. Graph G is said to be k−γc−critical if γc(G) = k but γc(G+e) < k for eac...

A graph G is 3-domination critical if its domination number γ is 3 and the addition of any edge decreases γ by 1. Let G be a 3-domination critical graph with toughness more than one. It was proved G is Hamiltonconnected for the cases α ≤ δ (Discrete Mathematics 271 (2003) 1-12) and α = δ + 2 (European Journal of Combinatorics 23(2002) 777-784). In this paper, we show G is Hamilton-connected for...

Given graph G = (V,E), a dominating set S is a subset of vertex set V such that any vertex not in S is adjacent to at least one vertex in S. The domination number of a graph G is the minimum size of the dominating sets of G. In this paper we study some results on domination number, connected, independent, total and restrained domination number denoted by γ(G), γc(G) ,γi(G), γt(G) and γr(G) resp...

For a connected graph G, a set of vertices W in G is called a Steiner dominating set if W is both a Steiner set and a dominating set. The minimum cardinality of a Steiner dominating set of G is its Steiner domination numberand is denoted by ) (G s  . In this paper, it is studied that how the Steiner domination number is affected by adding a single edge to paths, complete graphs, cycles, star a...