A dominating set of a graph is a set S of vertices such that every vertex in the graph is either in S or is adjacent to a vertex in S. The domination number of a graph G, denoted γ (G), is theminimum cardinality of a dominating set ofG. We show that ifG is an n-vertexmaximal outerplanar graph, then γ (G) ≤ (n + t)/4, where t is the number of vertices of degree 2 in G. We show that this bound is...

Definition of dominating function on a fractional graph G has been introduced. Fractional parameters such as domination number and upper defined. Domination with fuzzy Intuitionistic environment, have found by formulating Linear Programming Problem.

Wedesign fast exponential time algorithms for some intractable graph-theoretic problems. Ourmain result states that aminimum optional dominating set in a graph of order n can be found in time O∗(1.8899n). Ourmethods to obtain this result involvematching techniques. The list of the considered problems includes Minimum Maximal Matching, 3Colourability, Minimum Dominating Edge Set, Minimum Connect...

Abstract: A 2-rainbow domination function of a graph G = (V, E) is a function f mapping each vertex v to a subset of {1, 2} such that ⋃ u∈N(v) f (u) = {1, 2} when f (v) = �, where N(v) is the open neighborhood of v. The weight of f is denoted by wf (G) = ∑ v∈V �f (v)�. The 2-rainbow domination number, denoted by r2(G), is the smallest wf (G) among all 2-rainbow domination functions f of G. The ...

A Roman dominating function on a graphG is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u ∈ V (G) for which f(u) = 0 is adjacent to at least one vertex v ∈ V (G) for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The Roman domination number γR(G) of G is the minimum weight of a Roman dominating function on G. A Ro...

A Roman dominating function (RDF) on a digraph $D$ is a function $f: V(D)rightarrow {0,1,2}$ satisfying the condition that every vertex $v$ with $f(v)=0$ has an in-neighbor $u$ with $f(u)=2$. The weight of an RDF $f$ is the value $sum_{vin V(D)}f(v)$. The Roman domination number of a digraph $D$ is the minimum weight of an RDF on $D$. A set ${f_1,f_2,dots,f_d}$ of Roman dominating functions on ...

We investigate in this paper the set of kadditive capacities dominating a given capacity, which we call the k-additive core. We study its structure through achievable families, which play the role of maximal chains in the classical case (k = 1), and show that associated capacities are elements (possibly a vertex) of the k-additive core when the capacity is (k+1)-monotone. As a particular case, ...

A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_...