Efficient domination in circulant graphs

2-Tuple Total Domination Number in Circulant Graphs

In this paper, a necessary and sufficient condition for the existence of an efficient 2-dominating set in a class of circulant graphs has been obtained and for those circulant graphs, an upper bound for the 2domination number is also obtained. For the circulant graphs Cir(n,A), where A = {1, 2, . . . , x, n − 1, n − 2, . . . , n − x} and x ≤ bn−1 2 c, the perfect 2-tuple total domination number...

Hypo-efficient domination and hypo-unique domination

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...

The bondage numbers and efficient dominations of vertex-transitive graphs

The bondage number of a graph G is the minimum number of edges whose removal results in a graph with larger domination number.A dominating setD is called an efficient dominating set ofG if |N−[v]∩D|=1 for every vertex v ∈ V (G). In this paper we establish a tight lower bound for the bondage number of a vertex-transitive graph. We also obtain upper bounds for regular graphs by investigating the ...

Domination in Circulant Graphs

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we determine the domination and the total domination number in the Circulant graphs Cn〈1, 3〉, and then study γ-criticality...

Vertex Domination Critical in Circulant Graphs