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...

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...

Define a Roman dominating function (RDF) of a graph G to be a function f : V (G) → {0, 1, 2} such that every u with f(u) = 0 has a neighbor v with f(v) = 2. The weight of f , w(f), is ∑ v∈V (G) f(v). The Roman domination number of G, γR(G), is the minimum weight of an RDF of G. It is easy to see that γ(G) ≤ γR(G) ≤ 2γ(G), where γ(G) is the domination number of G. In this paper, we determine pro...

Abstract. The middle edge dominating graph Med(G) of a graph G=(V,E) is a graph with the vertex set E ∪S where S is the set of all minimal edge dominating set G and with two vertices u, v є E ∪S adjacent if u є E and V=F is a minimal edge dominating set of G containing u or u,v are not disjoint minimal edge dominating sets in S .In this paper we discuss about the middle edge dominating graph of...

Given a simple undirected graph, the minimum connected dominating set problem is to find a minimum cardinality subset of vertices D inducing a connected subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual routing backbone in wireless networks. This paper first proposes a constant-ratio...

1. Any vertex that is incident to an observed edge is observed. 2. Any edge joining two observed vertices is observed. The power domination problem is a variant of the classical domination problem in graphs and is defined as follows. Given an undirected graph G = (V, E), the problem is to find a minimum vertex set SP ⊆ V , called the power dominating set of G, such that all vertices in G are ob...

A signed graph is a graph in which every edge is designated to be either positive or negative; it is balanced if every cycle contains an even number of negative edges. A marked signed graph is a signed graph each vertex of which is designated to be positive or negative, and it is consistent if every cycle in the signed graph possesses an even number of negative vertices. Signed line graph L(S) ...

Results about wreath products of circulant graphs are used to construct infinitely many circulant graphswith efficient dominating setswhose elements need not be equally spaced in Zn. It is proved that if a circulant graph of large degree has an efficient dominating set, then either its elements are equally spaced, or the graph is the wreath product of a smaller circulant graph with an efficient...