A dominating set $S$ of $G$ is an \textit{equitable dominating set} of $G$ if for every $v \in V(G) \setminus S$, there exists $u \in S$ such that $uv \in V(G)$ and $\displaystyle{\left|\deg(u) - \deg(v)\right| \leq 1.}$ A dominating set $S$ of $G$ is a \textit{rings dominating set} of $G$ if every vertex $v \in V(G) \setminus S$ is adjacent to atleast two vertices $V(G) \setminus S$. In this paper, we examine the conditions at which the equitable dominating set and the rings dominating set coincide, and thus naming the dominating set as \textit{equitable rings dominating set}. The minimum cardinality of an equitable rings dominating set of a graph $G$ is called the \textit{equitable rings domination number} of $G$, and is denoted by $\gamma_{eri}(G)$. Moreover, we examine determine the equitable rings domination number of many graphs, and graphs formed by some binary operations.

A dominating set $S$ of $G$ is an \textit{equitable dominating set} of $G$ if for every $v \in V(G) \setminus S$, there exists $u \in S$ such that $uv \in V(G)$ and $\displaystyle{\left|\deg(u) - \deg(v)\right| \leq 1.}$ A dominating set $S$ of $G$ is a \textit{rings dominating set} of $G$ if every vertex $v \in V(G) \setminus S$ is adjacent to atleast two vertices $V(G) \setminus S$. In this paper, we examine the conditions at which the equitable dominating set and the rings dominating set coincide, and thus naming the dominating set as \textit{equitable rings dominating set}. The minimum cardinality of an equitable rings dominating set of a graph $G$ is called the \textit{equitable rings domination number} of $G$, and is denoted by $\gamma_{eri}(G)$. Moreover, we examine determine the equitable rings domination number of many graphs, and graphs formed by some binary operations.

An equitable domination has interesting application in the context of social networks. In a network, nodes with nearly equal capacity may interact with each other in a better way. In the society persons with nearly equal status, tend to be friendly. In this paper, we introduce new variant of equitable domination of a graph. Basic properties and some interesting results have been obtained.

A total Roman dominating function on a graph G is a function f: V (G)! f0; 1; 2g such that for every vertex v 2 V (G) with f(v) = 0 there exists a vertex u 2 V (G) adjacent to v with f(u) = 2, and the subgraph induced by the set fx 2 V (G): f(x)  1g has no isolated vertices. The total Roman domination number of G, denoted tR(G), is the minimum weight! (f) = P v2V (G) f(v) among all total Roman dominating functions f on G. It is known that tR(G)  t2(G) + (G) for any graph G with neither isolated vertex nor components isomorphic to K2, where t2(G) and (G) represent the semitotal domination number and the classical domination number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.

In this paper, we investigate domination number as well as signed domination numbers of Cay(G: S) for all cyclic group G of order n, where n ϵ {pm, pq} and S = {k < n: gcd(k, n) = 1}. We also introduce some families of connected regular graphs 􀀀 such that S (􀀀 ) ϵ {2, 3, 4, 5}.

Freshwater lens on small islands may easily be overexploited or polluted due to overdrafts of fresh water by pumping which causes drawdown of the water table a rise or up-coning of the saltwater interface. Present study is concerned with using a three-dimensional finite-difference numerical model to simulate the groundwater flow and transport model to predict the behavior of groundwater system in Manukan Island. The simulations were done using variable density SEAWAT-2000 for three selected pumping schemes. Hydraulic heads (groundwater level) is the highest at the center of the island and decreases in radial shape towards the coast in all the pumping schemes (PS1-PS3). The chloride concentration in the studied aquifer increased by 98.7% in the pumping well if the pumping rate is doubled by the current (PS2 to PS3). The 1.4% seawater-freshwater mixing moves further forward to inland about 1.6m when the current pumping rate was doubled whereas moves backward to sea about 1.7m if the current pumping rate is reduced by 50%. This preliminary model of Manukan island aquifer shows that an overexploitation of groundwater in Manukan Island contributes to the seawater intrusion. Adjusting the future groundwater pumping scheme and improving groundwater management strategies are necessary to protect the freshwater aquifers. The current numerical model is a reasonable representation of the aquifer in Small Island which can be used in similar small islands with similar hydrogeological conditions in elsewhere.

Let G = (V, E) be a simple, undirected and connected graph. A Roman dominating function (RDF) on the graph G is a function f: V → {0, 1, 2} such that each vertex v ∈ V with f(v) = 0 is adjacent to at least one vertex u ∈ V with f(u) = 2. A total Roman dominating function (TRDF) of G is a function f: V → {0, 1, 2} such that (i) it is a Roman dominating function, and (ii) the vertices with non-zero weights induce a subgraph with no isolated vertex. The total Roman dominating set (TRDS) problem is to minimize the associated weight, f(V ) = P u∈V f(u), called the total Roman domination number (γtR(G)). Similarly, a subset S ⊆ V is said to be a total dominating set (TDS) on the graph G if (i) S is a dominating set of G, and (ii) the induced subgraph G[S] does not have any isolated vertex. The objective of the TDS problem is to minimize the cardinality of the TDS of a given graph. The TDS problem is NP-complete for general graphs. In this paper, we propose a simple 10. 5-factor approximation algorithm for TRDS problem in UDGs. The running time of the proposed algorithm is O(|V | log k), where k is the number of vertices with weights 2. It is an improvement over the best-known 12-factor approximation algorithm with running time O(|V | log k) available in the literature. Next, we propose another algorithm for the TDS problem in UDGs, which improves the previously best-known approximation factor from 8 to 7. 79. The running time of the proposed algorithm is O(|V | + |E|).

A defective vertex coloring of a graph is a coloring in which some adjacent vertices may have the same color. An edge whose adjacent vertices have the same color is called a bad edge. A defective coloring of a graph $G$ with minimum possible number of bad edges in $G$ is known as a near proper coloring of $G$.  In this paper, we introduce the notion of equitable near proper coloring of graphs and determine the minimum number of bad edges obtained from an equitable near proper coloring of some graph classes.

This paper, presents a DEA-based method for allocating a shared costs to Decision Making Units which Cook and Kress proposed in their paper [European Journal of Operational Research 119 (1999)]. To illustrate the method, numerical result for an example from the literature is presented.