A Roman dominating function of a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(...

A 2-rainbow dominating function ( ) of a graph  is a function  from the vertex set  to the set of all subsets of the set  such that for any vertex  with  the condition  is fulfilled, where  is the open neighborhood of . A maximal 2-rainbow dominating function on a graph  is a 2-rainbow dominating function  such that the set is not a dominating set of . The weight of a maximal    is the value . ...

this paper attempts to examine the physical and financial performance of some primary dairy co-operatives in west bengal state in indian context. the study suggests that financial performance indicators dominate over physical ones, and all the dominating variables have long term positive impact on primary milk producers’ co-operative societies (pmpcss). also, the significant impact of financial...

This paper is devoted to the study of quadruple Roman domination in trees, and it a contribution Special Issue “Theoretical computer science discrete mathematics” Symmetry. For any positive integer k, [k]-Roman dominating function ([k]-RDF) simple graph G from vertex set V {0,1,2,…,k+1} if for u?V with f(u)<k, ?x?N(u)?{u}f(x)?|{x?N(u):f(x)?1}|+k, where N(u) open neighborhood u. The weight [k...

Let G = (V,E) be a graph and let f be a function f : E → {0, 1, 2}. An edge x with f(x) = 0 is said to be undefended with respect to f if it is not incident to an edge with positive weight. The function f is a weak edge Roman dominating function (WERDF) if each edge x with f(x) = 0 is incident to an edge y with f(y) > 0 such that the function f ′ : E → {0, 1, 2}, defined by f ′(x) = 1, f ′(y) =...

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...

Dominating sets in their many variations model a wealth of optimization problems like facility location or distributed le sharing. For instance, when a request can occur at any node in a graph and requires a server at that node, a minimumdominating set represents a minimum set of servers that serve an arbitrary single request by moving a server along at most one edge. This paper studies dominat...

Let G = (V,E) be a graph and f be a function f : V → {0, 1, 2}. A vertex u with f(u) = 0 is said to be undefended with respect to f , if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f ′ : V → {0, 1, 2} defined by f ′ (u) = 1, f ′ (v) = f...

We investigate a domination-like problem from the exact exponential algorithms viewpoint. The classical Dominating Set problem ranges among one of the most famous and studied NP -complete covering problems [6]. In particular, the trivial enumeration algorithm of runtime O∗(2n) 4 has been improved to O∗(1.4864n) in polynomial space, and O∗(1.4689n) with exponential space [9]. Many variants of th...