For a positive integer k, a total {k}-dominating function of a digraph D is a function f from the vertex set V (D) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (D), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) consists of all vertices of D from which arcs go into v. A set {f1, f2, . . . , fd} of total {k}-dominating functions of D with the property that ∑ d i=1 fi(...

‎a set $s$ of vertices of a graph $g=(v,e)$ without isolated vertex‎ ‎is a {em total dominating set} if every vertex of $v(g)$ is‎ ‎adjacent to some vertex in $s$‎. ‎the {em total domatic number} of‎ ‎a graph $g$ is the maximum number of total dominating sets into‎ ‎which the vertex set of $g$ can be partitioned‎. ‎we show that the‎ ‎total domatic number of a random $r$-regular graph is almost‎...

A restrained Roman dominating function (RRD-function) on a graph \(G=(V,E)\) is \(f\) from \(V\) into \(\{0,1,2\}\) satisfying: (i) every vertex \(u\) with \(f(u)=0\) adjacent to \(v\) \(f(v)=2\); (ii) the subgraph induced by vertices assigned 0 under has no isolated vertices. The weight of an RRD-function sum its value over whole set vertices, and domination number minimum \(G.\) In this paper...

Given a graph G together with a capacity function c : V (G) → N, we call S ⊆ V (G) a capacitated dominating set if there exists a mapping f : (V (G) \ S) → S which maps every vertex in (V (G) \S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v ∈ S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a ca...

<abstract><p>Let $ G be a graph with vertex set V(G) $. A function f:V(G)\rightarrow \{0, 1, 2\} is Roman dominating on if every v\in for which f(v) = 0 adjacent to at least one u\in such that f(u) 2 The domination number of the minimum weight \omega(f) \sum_{x\in V(G)}f(x) among all functions f In this article we study direct product graphs and rooted graphs. Specifically, give sev...

Let G be a graph with no isolated vertex and let N(v) the open neighbourhood of v∈V(G). f:V(G)→{0,1,2} function Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is strongly total Roman dominating on if subgraph induced by V1∪V2 has N(v)∩V2≠∅ v∈V(G)\V2. The domination number G, denoted γtRs(G), defined as minimum weight ω(f)=∑x∈V(G)f(x) among all functions G. This paper devoted to study it ...