Abstract:
Let G = (V, E) be a simple graph. A set SíV is a dominating set of G, if every vertex in V-S is adjacent to atleast one vertex in S. Letbe the square of the Pathand letdenote the family of all dominating sets ofwith cardinality i. Let. In this paper, we obtain a recursive formula for. Using this recursive formula, we construct the polynomial, , which we call domination polynomial ofand obtain some properties of this polynomial.

Abstract:
A dominating set D in a graph G is called
an injective equitable dominating set (Inj-equitable dominating set) if for
every , there exists such that u is adjacent to v and . The minimum cardinality
of such a dominating set is denoted by and is called the Inj-equitable
domination number of G. In this paper, we introduce the injective equitable
domination of a graph and study its relation with other domination parameters.
The minimal injective equitable dominating set, the injective equitable
independence number , and the injective equitable domatic number are defined.

Gravier et
al. established bounds on the size of a minimal totally dominant subset for graphs P_{k}□_{}P_{m}. This paper offers an
alternative calculation, based on the following lemma: Let so k≥3 and r≥2. Let H be an r-regular finite graph, and put G=P_{k}_{}□H. 1) If a perfect totally dominant subset
exists for G, then it is minimal; 2) If r＞2 and a perfect totally dominant subset exists
for G, then every minimal totally
dominant subset of G must be perfect. Perfect dominant subsets
exist forP_{k}_{}□_{ }C_{n} when k and n satisfy specific modular conditions. Bounds
for r_{t}(P_{k}

Abstract:
In this paper, sharp upper bounds for the domination number, total domination number and connected domination number for the Cayley graph G = Cay(D2n, Ω) constructed on the finite dihedral group D2n, and a specified generating set Ω of D2n. Further efficient dominating sets in G = Cay(D2n, Ω) are also obtained. More specifically, it is proved that some of the proper subgroups of D2n are efficient domination sets. Using this, an E-chain of Cayley graphs on the dihedral group is also constructed.

Abstract:
A set $D$ of vertices in a graph $G$ is a locating-dominating set if for every two vertices $u, v$ of $G \setminus D$ the sets $N(u) \cap D$ and $N(v) \cap D$ are non-empty and different. In this paper, we characterize vertices that are in all or in no minimum locating dominating sets in trees. The characterization guarantees that the $\gamma_L$-excellent tree can be recognized in a polynomial time.

Abstract:
The domination integrity of a connected graph G= (V(G), E(G)) is denoted as DI(G) and defined by DI(G) = min{*S*+ m(G-S) : S is a dominating set } where m(G-S) is the order of a maximum component of G-S . We discuss domination integrity in the context of some graph operations like duplication of an edge by vertex and duplication of vertex by an edge.

Abstract:
Generalized Petersen graphs are an important class of commonly used interconnection networks and have been studied . The total domination number of generalized Petersen graphs P(m,2) is obtained in this paper.

Abstract:
Let G = (V, E) be a simple graph. A set SE(G) is an edge-vertex
dominating set of G (or simply an ev-dominating set), if for all vertices v V(G); there exists an
edge eS such that e dominates v. Let denote the family of all ev-dominating sets of with cardinality i. Let . In this paper, we
obtain a recursive formula for . Using this
recursive formula, we construct the polynomial, , which we call edge-vertex domination polynomial of (or simply an ev-domination polynomial of ) and obtain some
properties of this polynomial.

Abstract:
In a graph G, a vertex dominates itself and its neighbours. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number is the minimum cardinality of a dominating set. A set is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of a double dominating set is called double domination number of G and is denoted by dd(G). The connectivity of a connected graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. In this paper we find an upper bound for the sum of the double domination number and connectivity of a graph and characterize the corresponding extremal graphs.

Abstract:
We consider and we study a general concept of domination for controlled and observed distributed systems. We give characterization results and the main properties of this notion for controlled systems, with respect to an output operator. We also examine the case of actuators and sensors. Various other situations are considered and applications are given. Then, we extend this study by comparing observed systems with respect to a control operator. Finally, we study the relationship between the notion of domination and the compensation one, in the exact and weak cases.