In a graph G=(V,E), where every vertex is assigned 0, 1 or 2, f an assignment such that 0 has at least one neighbor 2 and all vertices labeled by are independent, then called outer independent Roman dominating function (OIRDF). The domination strengthened if 1, 3, each two neighbors double (OIDRDF). weight of (OIDRDF) OIRDF the sum f(v) for v?V. (double) number (?oidR(G)) ?oiR(G) minimum taken ...

Let G be a connected graph. A function f : V (G) → {0, 1, 2, 3} is double Roman dominating of if for each v ∈ with f(v) = 0, has two adjacent vertices u and w which f(u) f(w) 2 or an vertex 3, to either 3. The minimum weight ωG(f) P v∈V the domination number G. In this paper, we continue study introduced studied by R.A. Beeler et al. in [2]. First, characterize some numbers small values terms 2...

In this paper, we define a new domination-like invariant of graphs. Let $${\mathbb {R}}^{+}$$ be the set non-negative numbers. $$c\in {\mathbb {R}}^{+}-\{0\}$$ number, and let G graph. A function $$f:V(G)\rightarrow is c-self-dominating if for every $$u\in V(G)$$ , $$f(u)\ge c$$ or $$\max \{f(v):v\in N_{G}(u)\}\ge 1$$ . The c-self-domination number $$\gamma ^{c}(G)$$ defined as ^{c}(G):=\min \{...

An outer-independent double Italian dominating function (OIDIDF)on a graph $G$ with vertex set $V(G)$ is a function$f:V(G)longrightarrow {0,1,2,3}$ such that if $f(v)in{0,1}$ for a vertex $vin V(G)$ then $sum_{uin N[v]}f(u)geq3$,and the set $ {uin V(G)|f(u)=0}$ is independent. The weight ofan OIDIDF $f$ is the value $w(f)=sum_{vin V(G)}f(v)$. Theminimum weight of an OIDIDF on a graph $G$ is cal...

Let G be a graph with no isolated vertex and f : V ( ) → {0, 1, 2} function. i = { x ∈ } for every . We say that is total Roman dominating function on if in 0 adjacent to at least one 2 the subgraph induced by 1 ∪ has vertex. The weight of ω ∑ v minimum among all functions domination number , denoted γ t R It known general problem computing NP-hard. In this paper, we show H nontrivial graph, th...

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)=0 adjacent to at least one assigned 3 or two vertices 2, and with f(u)=1 2 3. The weight of f equals w(f)=?v?Vf(v). domination number ?dR(G) G minimum G. We obtain closed expressions generalized Petersen graphs P(5k,k). It proven ?dR(P(5k,k))=8k k?2,3mod5 8k??dR(P(5...