On Graphs with Equal Connected Domination and 2-connected Domination Numbers

A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.

The Generalization of Signed Domination Number of Two Classes of Graphs

Let be a graph. A function is said to be a Signed Dominating Function (SDF) if holds for all . The signed domination number . In this paper, we determine the exact value of the Signed Domination Number of graphs and for , which is generalized the known results, respectively, where and are denotes the k-th power graphs of cycle and path .

Total Domination number of Generalized Petersen Graphs

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.

On the 2-Domination Number of Complete Grid Graphs

A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].

An Upper Bound for Total Domination Number

Let G=(V, E) be a simple graph without an isolate. A subset T of V is a total dominating set of G if for any there exists at least one vertex such that .The total domination number γ1(G) of G is the minimum order of a total dominating set of G. This paper proves that if G is a connected graph with n≥3 vertices and minimum degree at least two.

On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number

Let G = （V, E） be a simple graph. A function f ： E → {＋1,-1} is called a signed cycle domination function （SCDF） of G if ∑e∈E（C） f（e） ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc（G） = min{∑e∈E f（e）｜ f is an SCDF of G}. This paper will characterize all maxima] planar graphs G with order n ≥ 6 and γ′sc（G） =n.

Characterization of Connected Graphs with Maximum Domination Number

Let G be a connected graph of order p, and let γ7(G) denote the domination number of G. Clearly, γ(G) ≤[p/2]. The aim of this paper is to characterize the graphs G that reaches this upper bound. The main results are as follows: (1) when p is even, γ(G) = p/2 if and only if either G C4 or G is the crown of a connected graph with p/2 vertices; (2) when p is odd, γ(G) = (p-1)/2 if and only if every spanning tree of G is one of the two classes of trees shown in Theorem 3.1.

On the Signed Domination Number of the Cartesian Product of Two Directed Cycles

Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, ?(mod 3) and bounds for otherwise.

On the Maximum Number of Dominating Classes in Graph Coloring

We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].

Bounds on Fractional Domination of Some Products of Graphs

Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n×P m (grid graph) with small n and m . But for large n and m , it is difficult to decide the exact fractional domination number. Motivated by this, nearly sharp upper and lower bounds are given to the fractional domination number of grid graphs. Furthermore, upper and lower bounds on the fractional total domination number of strong direct product of graphs are given.

Bondage and Reinforcement Number of γ_f for Complete Multipartite Graph

The bondage number of γ f, b f(G) , is defined to be the minimum cardinality of a set of edges whose removal from G results in a graph G′ satisfying γ f(G′)> γ f(G) . The reinforcement number of γ f, r f(G) , is defined to be the minimum cardinality of a set of edges which when added to G results in a graph G′ satisfying γ f(G′)< γ f(G) . G.S.Domke and R.C.Laskar initiated the study of them and gave exact values of b f(G) and r f(G) for some classes of graphs. Exact values of b f(G) and r f(G) for complete multipartite graphs are given and some results are extended.

On Signed Domination of Grid Graph

Let G(V, E) be a finite connected simple graph with vertex set V(G). A function is a signed dominating function f : V(G)→{−1,1} if for every vertex v ∈ V(G), the sum of closed neighborhood weights of v is greater or equal to 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 6, 7 and arbitrary n.

A Lower Bound for the Algebraic Connectivity of a Graph in Terms of the Domination Number

We investigate how the algebraic connectivity of a graph changes by relocating a connected branch from one vertex to another vertex, and then minimize the algebraic connectivity among all connected graphs of order n with fixed domination number γ≤（n＋2）/3, and finally present a lower bound for the algebraic connectivity in terms of the domination number. We also characterize the minimum algebraic connectivity of graphs with domination number half their order.

Bondage Number of 1-Planar Graph

The bondage number of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph a domination number greater than the domination number of G. In this paper, we prove that for a 1-planar graph G.

A Note on Edge-Domsaturation Number of a Graph

The edge-domsaturation number ds'(G) of a graph G = (V, E) is the least positive integer k such that every edge of G lies in an edge dominating set of cardinality k. In this paper, we characterize unicyclic graphs G with ds'(G) = q – Δ'(G) + 1 and investigate well-edge dominated graphs. We further define γ'–-critical, γ'+-critical, ds'–-critical, ds'+-critical edges and study some of their properties.

Domination and Eternal Domination of Jahangir Graph

In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. We consider the “all guards move” of the eternal dominating set problem. In which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current position after each attack. If the new formed set of guards is still a dominating set of the graph then we successfully defended the attack. Our goal is to find the minimum number of guards required to eternally protect the graph. We call this number the m-eternal domination number and we denote it by . In this paper we find the eternal domination number of Jahangir graph Js,m for s=2,3 and arbitrary m. We also find the domination number for J3,m .

The Middle Equitable Dominating Graphs

Let G= (V, E) be a graph and A(G) is the collection of all minimal equitable dominating set of G. The middle equitable dominating graph of G is the graph denoted by Med(G) with vertex set the disjoint union of V∪A(G) and (u, v) is an edge if and only if u ∩ v ≠ φ whenever u, v ∈ A(G) or u ∈ v whenever u ∈ v and v ∈ A(G) . In this paper, characterizations are given for graphs whose middle equitable dominating graph is connected and Kp∈Med(G) . Other properties of middle equitable dominating graphs are also obtained.

On the dominating set in circular arc graphs

圆弧图是比区间图更广泛的一类交图.设 G=(V,E)是任意图,用V(G)、i(G)、ir(G)和β(G)分别表示 G 的控制数、独立控制数、无赘数和点覆盖数.讨论了圆弧图的这些图参数之间的关系.

Domination Number in Graphs with Minimum Degree Two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.

Some New Results on Double Domination in Graphs

Each vertex of a graph G = （V, E） is said to dominate every vertex in its closed neighborhood. A set S C V is a double dominating set for G if each vertex in V is dominated by at least two vertices in S. The smallest cardinality of a double dominating set is called the double dominating number dd（G）. In this paper, new relationships between dd（G） and other domination parameters are explored and some results of [1] are extended. Furthermore, we give the Nordhaus-Gaddum-type results for double dominating number.