A subset S ⊆ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number dd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision number sddd(G) is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the double domination n...

A graph G is 3-domination critical if its domination number γ is 3 and the addition of any edge decreases γ by 1. It was proved by Favaron et al. that α ≤ δ + 2 for any connected 3-domination critical graph. Denote by τ (G) the toughness of a graph G . Recently Chen et al. conjectured that a connected 3-domination critical graph G is Hamilton-connected if and only if τ (G) > 1 and showed the co...

If G=(VG,EG) is a graph of order n, we call S⊆VG an isolating set if the induced by VG−NG[S] contains no edges. The minimum cardinality G called isolation number G, and it denoted ι(G). It known that ι(G)≤n3 bound sharp. A subset dominating in NG[S]=VG. domination number, γ(G). In this paper, analyze family trees T where ι(T)=γ(T), prove ι(T)=n3 implies ι(T)=γ(T). Moreover, give different equiv...

A dominating set D of G is called a split dominating set of G if the subgraph induced by the subset V (G) − D is disconnected. The cardinality of a minimum split dominating set is called the minimum split domination number of G. Such subset and such number was introduced in [4]. In [2], [3] the authors estimated the domination number of products of graphs. More precisely, they were study produc...

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...

In this paper, sharp upper bounds for the domination number, total domination number and connected domination number for the Cayley graph G = Cay(D2n, Ω) constructed on the finite dihedral group D2n, and a specified generating set Ω of D2n. Further efficient dominating sets in G = Cay(D2n, Ω) are also obtained. More specifically, it is proved that some of the proper subgroups of D2n are efficie...