Recent articles by ReVelle [20, 21] in the Johns Hopkins Magazines suggested a new variation of domination called Roman domination, see also [22] for an integer programming formulation of the problem. Since then, there have been several articles on Roman domination and its variations [2, 3, 4, 5, 6, 11, 12, 14, 15, 16, 18, 24, 23, 25]. Emperor Constantine had the requirement that an army or leg...

a dominating set $d subseteq v$ of a graph $g = (v,e)$ is said to be a connected cototal dominating set if $langle d rangle$ is connected and $langle v-d rangle neq phi$, contains no isolated vertices. a connected cototal dominating set is said to be minimal if no proper subset of $d$ is connected cototal dominating set. the connected cototal domination number $gamma_{ccl}(g)$ of $g$ is the min...

A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3}, satisfying the condition that every vertex u for which f(u)=1 adjacent to at least one assigned 2 or 3, and with f(u)=0 3 two vertices 2. The weight of f equals sum w(f)=?v?Vf(v). minimum G called domination number ?dR(G) G. We obtain tight bounds in some cases closed expressions generalized Petersen graphs P(ck,k). In shor...

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 Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V rightarrow {0, 1, 2}$ suchthat every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Sigma_{vin V} f(v)$The Roman domination number, $gamma_R(G)$, of $G$ is theminimum weight of an RDF on $G$.An RDF of minimum weight is called a $gamma_R$-function.A graph G is said to be $g...

In a graph, a vertex is said to dominate itself and all its neighbors. A double dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. The double domination number γ×2(G) is the minimum cardinality of a double dominating set of G. A graph G without isolated vertices is called edge removal critical with respect to double domination, or just γ×2-criti...

For a graph [Formula: see text], double Roman dominating function (DRDF) is text] having the property that if for some vertex then has at least two neighbors assigned under or one neighbor with and text]. The weight of DRDF sum minimum on domination number denoted by bondage cardinality among all edge subsets such In this paper, we study in graphs. We determine several families graphs, present ...