In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, trees, DAGs, cycles, and bipartite graphs. We also provide the idomatic number for special cases of some of these families of digraphs.

Let G be a simple graph. An independent set is a set of pairwise non-adjacent vertices. The number of vertices in a maximum independent set of G is denoted by a (G). In this paper, we characterize graphs G with n vertices and with maximum number of maximum independent sets provided that a (G) £2 or a (G) ³n-3.

The unit disk graph is used to model a wireless sensor network when all sensors have the same communication radius. Hence, the following two optimization problems have been investigated by the researchers: maximal independent set in the network and minimal network dominating set. Since these problems are Np-hard, several algorithms have been presented for their approximation. In this paper, we have presented a honeycomb graph and algorithmic matrix methods for approximating the maximal independent set in the network. Finally, we have confirmed the validity of the algorithm and its complexity and studied it with a numerical example.

A coalition in a graph G = (V, E) consists of two disjoint sets V1 and V2 of vertices, such that neither V1 nor V2 is a dominating set, but the union V1 , V2 is a dominating set of G. A coalition partition in a graph G of order n = |V| is a vertex partition π,= {V1, V2, …, , Vk} such that every set Vi either is a dominating set consisting of a single vertex of degree n-1, or is not a dominating set but forms a coalition with another set Vj. Associated with every coalition partition π,of a graph G is a graph called the coalition graph of G with respect to π, , denoted CG(G,π, ), the vertices of which correspond one-to-one with the sets V1,V2,…, , Vk of π,and two vertices are adjacent in CG(G,π,) if and only if their corresponding sets in π,form a coalition. In this paper, we initiate the study of coalition graphs and we show that every graph is a coalition graph.

The Production Possibility set (PPS) is defined as the set of all inputs and outputs of a system in which inputs can produce outputs. In Data Envelopment Analysis (DEA), it is highly important to identify the defining hyperplanes and especially the strong defining hyperplanes of the empirical PPS. Although DEA models can determine the efficiency of a Decision Making Unit (DMU), but they cannot present efficient frontiers of PPS completely. The notion of defining hyperplanes is crucial to marginal discussions, marginal rates, marginal rates of substitution, sensitivity analysis, returns to scale, and in particular, the efficiency analysis of DMUs. In this paper, we propose a new method to determine all strong efficient(Pareto-efficient) DMUs and strong defining hyperplanes of the PPS with variable returns to scale which strong efficient DMUs are located on them. Furthermore, we apply the proposed method to find the normal vectors or gradient of the strong defining hyperplanes of the PPS including strong efficient DMUs. Consequently, the equations of these hyperplanes are determined. To illustrate the ability of the proposed method, some numerical examples are finally provided. Our method can be easily implemented using existing packages for operation research, such as GAMS.

‎Let $r\geq 2$. A subset $S$ of vertices of a graph $G$ is a $r$-hop independent dominating set if every vertex outside $S$ is at distance $r$ from a vertex of $S$, and for any pair $v, w\in S$, $d(v, w)\neq r$. A $r$-hop Roman dominating function ($r$HRDF) is a function $f$ on $V(G)$ with values $0,1$ and $2$ having the property that for every vertex $v \in V$ with $f(v) = 0$ there is a vertex $u$ with $f(u)=2$ and $d(u,v)=r$. A $r$-step Roman dominating function ($r$SRDF) is a function $f$ on $V(G)$ with values $0,1$ and $2$ having the property that for every vertex $v$ with $f(v)=0$ or $2$, there is a vertex $u$ with $f(u)=2$ and $d(u,v)=r$. A $r$HRDF $f$ is a $r$-hop Roman independent dominating function if for any pair $v, w$ with non-zero labels under $f$, $d(v, w)\neq r$. We show that the decision problem associated with each of $r$-hop independent domination, $r$-hop Roman domination, $r$-hop Roman independent domination and $r$-step Roman domination is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs.

In this paper we present some properties of set-norm exhaustive set multi functions and also of atoms and pseudo-atoms of set multi functions taking values in the family of non-empty subsets of a commutative semi group with unity.

In this paper we define the Soft set Product as a product of many soft sets and afterwards we extend it to the HyperSoft set. Similarly, the IndetermSoft Product is extended to the IndetermHyperSoft set. We also present several applications of the Soft set Product to Fuzzy (and fuzzy-extensions)  Soft set Product and to IndetermSoft set and IndetermHyperSoft set.