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...

Efficient communication between nodes in ad-hoc networks can be established through repeated cluster formations with designated clusterheads. In this context minimum d-hop dominating set problem was introduced for cluster formation in ad-hoc networks and is proved to be NP-complete. Hence, an exact solution to this problem for certain subclass of graphs (representing an ad-hoc network) can be b...

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.

Given a graph G = ( V , E ) and an integer k ? 1 - hop dominating set D of is subset such that, for every vertex v ? there exists node u whose distance from at most . A -hop minimum cardinality called In this paper, we present linear-time algorithm that finds in cactus graphs which improves the O n 3 -time Borradaile Le (2017). To achieve this, show problem unicyclic reduces to piercing circula...

Let G = ( V, E ) be a simple graph with vertex setxs V and edge set . A mixed Roman dominating function of is f : ∪ → {0, 1, 2} satisfying the condition that every element x ∈ for which f(x) 0 adjacent or incident to at least one y f(y) 2. The weight ω( ∑ domination number γ R minimum We first show problem computing * NP-complete bipartite graphs then we present upper lower bounds on number, so...

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...

An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function$fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighborsassigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vi...

In a wireless network, messages need to be sent on in an optimized way to preserve the energy of the network. A minimum connected dominating set (MCDS) offers an optimized way of sending messages. However, MCDS construction is a NP-Hard problem. In this paper, we propose a new degree-based greedy approximation algorithm named as Connected Pseudo Dominating Set Using 2 Hop Information (CPDS2HI),...