Abstract:
Let $G = (V, E)$ be a connected graph. A set $S$ of vertices in $G$ is a perfect dominating set if every vertex $v$ in $V-S$ is adjacent to exactly one vertex in $S$. A perfect dominating set $S$ is said to be a neighborhood connected perfect dominating set (ncpd-set) if the induced subgraph $$ is connected. The minimum cardinality of a ncpd-set of $G$ is called the neighborhood connected perfect domination number of $G$ and is denoted by $gamma_{ncp}(G)$. In this paper we initiate a study of this parameter.

Abstract:
For a graph , a subset of is called an edge dominating set of if every edge not in is adjacent to some edge in . The edge domination number of is the minimum cardinality taken over all edge dominating sets of . Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles. 1. Introduction The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it because of its many and varied applications in such fields as linear algebra and optimization, design and analysis of communication networks, and social sciences and military surveillance. Many variants of dominating models are available in the existing literature. For a comprehensive bibliography of papers on the concept of domination, readers are referred to Hedetniemi and Laskar [1]. The present paper is focused on edge domination in graphs. We begin with simple, finite, connected, and undirected graph of order . The set of vertices in a graph is called a dominating set if every vertex is either an element of or is adjacent to an element of . A dominating set is a minimal dominating set (or MDS) if no proper subset is a dominating set. The minimum cardinality of a dominating set of is called the domination number of which is denoted by , and the corresponding dominating set is called a -set of . The open neighborhood of is the set of vertices adjacent to , and the set is the closed neighborhood of . For any real number , denotes the smallest integer not less than and denotes the greatest integer not greater than . An edge of a graph is said to be incident with the vertex if is an end vertex of . In this case, we also say that is incident with . Two edges and which are incident with a common vertex are said to be adjacent. In a graph , a vertex of degree one is called a pendant vertex, and an edge incident with a pendant vertex is called a pendant edge. A subset is an edge dominating set if each edge in is either in or is adjacent to an edge in . An edge dominating set is called a minimal edge dominating set (or MEDS) if no proper subset of is an edge dominating set. The edge domination number is the minimum cardinality among all minimal edge dominating sets. The concept of edge domination was introduced by Mitchell and Hedetniemi [2] and it is explored by many researchers. Arumugam and Velammal [3] have discussed the edge domination in graphs while the fractional edge domination in graphs is discussed in Arumugam and Jerry [4]. The complementary edge domination in graphs is studied by Kulli and

Abstract:
An edge Roman dominating function of a graph $G$ is a function $f\colon E(G) \rightarrow \{0,1,2\}$ satisfying the condition that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e')=2$. The edge Roman domination number of $G$, denoted by $\gamma'_R(G)$, is the minimum weight $w(f) = \sum_{e\in E(G)} f(e)$ of an edge Roman dominating function $f$ of $G$. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if $G$ is a graph of maximum degree $\Delta$ on $n$ vertices, then $\gamma_R'(G) \le \lceil \frac{\Delta}{\Delta+1} n \rceil$. While the counterexamples having the edge Roman domination numbers $\frac{2\Delta-2}{2\Delta-1} n$, we prove that $\frac{2\Delta-2}{2\Delta-1} n + \frac{2}{2\Delta-1}$ is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of $k$-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on $n$ vertices is at most $\frac{6}{7}n$, which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain $K_{2,3}$ as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs.

Abstract:
A dominating set $D \subset V(G)$ is a weakly connected dominating set in $G$ if the subgraph $G[D]_w = (N_{G}[D],E_w)$ weakly induced by $D$ is connected, where $E_w$ is the set of all edges with at least one vertex in $D$. The weakly connected domination number $\gamma_w(G)$ of a graph $G$ is the minimum cardinality among all weakly connected dominating sets in $G$. The graph is said to be weakly connected domination critical ($\gamma_w$-critical) if for each $u, v \in V(G)$ with $v$ not adjacent to $u$, $\gamma_w(G + vu) < \gamma_w (G)$. Further, $G$ is k-$\gamma_w$-critical if $\gamma_w(G) = k$ and for each edge $e \nin E(G)$, $\gamma_w(G + e) < k$. In this paper we consider weaklyconnected domination critical graphs and give some properties of 3-$\gamma_w$-critical graphs.

Abstract:
Let be a simple graph of order . The domination polynomial of is the polynomial , where is the number of dominating sets of of size . Every root of is called the domination root of . In this paper, we study the domination polynomial of some graph operations. 1. Introduction Let be a simple graph. For any vertex , the open neighborhood of is the set and the closed neighborhood is the set . For a set , the open neighborhood of is and the closed neighborhood of is . A set is a dominating set if , or equivalently, every vertex in is adjacent to at least one vertex in . An -subset of is a subset of of cardinality . Let be the family of dominating sets of which are -subsets and let . The polynomial is defined as domination polynomial of [1]. This polynomial has been introduced by the author in his Ph.D. thesis in 2009 [2]. A root of is called a domination root of . More recently, domination polynomial has found application in network reliability [3]. For more information and motivation of domination polynomial and domination roots refer to [1, 2]. The join of two graphs and with disjoint vertex sets and and edge sets and is the graph union together with all the edges joining and . The corona of two graphs and , is the graph formed from one copy of and copies of , where the th vertex of is adjacent to every vertex in the th copy of [4]. the Cartesian product of two graphs and is denoted by , is the graph with vertex set and edges between two vertices and if and only if either and or and . In this paper, we study the domination polynomials of some graph operations. 2. Main Results As is the case with other graph polynomials, such as chromatic polynomials and independence polynomials, it is natural to consider the domination polynomial of composition of two graphs. It is not hard to see that the formula for domination polynomial of join of two graphs is obtained as follows. Theorem 1 (see [1]). Let and be graphs of orders and , respectively. Then It is obvious that this operation of graphs is commutative. Using this product, one is able to construct a connected graph with the number of dominating sets , where is an arbitrary odd natural number; see [5]. Let to consider the corona of two graphs. The following theorem gives us the domination polynomial of graphs of the form which is the first result for domination polynomial of specific corona of two graphs. Theorem 2 (see [1]). Let be a graph. Then if and only if for some graph of order . It is easy to see that the corona operation of two graphs does not have the commutative property. The following theorem gives

Abstract:
As a generalization of connected domination in a graph G we consider domination by sets having at most k components. The order γ c k (G) of such a smallest set we relate to γ c (G), the order of a smallest connected dominating set. For a tree T we give bounds on γ c k (T) in terms of minimum valency and diameter. For trees the inequality γ c k (T)≤ n-k-1 is known to hold, we determine the class of trees, for which equality holds.

Abstract:
A dominating set in a graph G=(V(G),E(G)) is a set D of vertices such that every vertex in V(G) D has a neighbor in D. A connected dominating set of a graph G is a dominating set whose induce subgraph is connected. The connected domination number gamma_c(G) is the minimum number of vertices of a connected dominating set of G. A graph G is connected domination dot-critical (cdd-critical) if contracting any two adjacent vertices decreases gamma_c(G); and G is totally connected domination dot-critical (tcdd-critical) if contracting any two vertices decreases gamma_c(G). We provide characterizations of tcdd-critical graphs for the classes of block graphs, split graphs and unicyclic graphs and a characterization of cdd-critical cacti.

Abstract:
Let $\gamma'_s(G)$ be the signed edge domination number of G. In 2006, Xu conjectured that: for any $2$-connected graph G of order $ n (n \geq 2),$ $\gamma'_s(G)\geq 1$. In this article we show that this conjecture is not true. More precisely, we show that for any positive integer $m$, there exists an $m$-connected graph $G$ such that $ \gamma'_s(G)\leq -\frac{m}{6}|V(G)|.$ Also for every two natural numbers $m$ and $n$, we determine $\gamma'_s(K_{m,n})$, where $K_{m,n}$ is the complete bipartite graph with part sizes $m$ and $n$.

Abstract:
Let $G=(V,E)$ be a graph. A function $f:E ightarrow [0,1]$ iscalled an {it edge dominating function} if $sumlimits_{xin N[e]}f(x)geq 1$ for all $ein E(G),$where $N[e]$ is the closed neighbourhood of the edge $e.$ An edge dominating function $f$ is calledminimal (MEDF) if for all functions $g:E ightarrow [0,1]$ with $g Keywords edge dominating function --- edge irredundant function --- edge independent function