Weak and Strong Reinforcement Number For a Graph

Introducing the weak reinforcement number which is the minimum number of added edges to reduce the weak dominating number, and giving some boundary of this new parameter and trees

Closeness centrality in some splitting networks

A central issue in the analysis of complex networks is the assessment of their robustness and vulnerability. A variety of measures have been proposed in the literature to quantify the robustness of networks, and a number of graph-theoretic parameters have been used to derive formulas for calculating network reliability. \textit{Centrality} parameters play an important role in the field of network analysis. Numerous studies have proposed and analyzed several \textit{centrality} measures. We consider \textit{closeness centrality} which is defined as the total graph-theoretic distance to all other vertices in the graph. In this paper, closeness centrality of some splitting graphs is calculated, and exact values are obtained

On the average lower bondage number of a graph

The domination number is an important subject that it has become one of the most widely studied topics in graph theory, and also is the most often studied property of vulnerability of communication networks. The vulnerability value of a communication network shows the resistance of the network after the disruption of some centers or connection lines until a communication breakdown. Let G = (V(G),E(G)) be a simple graph. The bondage number b(G) of a nonempty graph G is the smallest number of edges whose removal from G result in a graph with domination number greater than that of G. If we think a graph as a modeling of network, the average lower bondage number of a graph is a new measure of the graph vulnerability and it is defined by bav(G) = 1/|E(G)| ∑e∈E(G)be(G), where the lower bondage number, denoted by be(G), of the graph G relative to e is the minimum cardinality of bondage set in G that contains the edge e. In this paper, the above mentioned new parameter has been defined and examined. Then upper bounds, lower bounds and exact formulas have been obtained for any graph G. Finally, the exact values have been determined for some well-known graph families

The Average Lower 2-Domination Number of Wheels Related Graphs and an Algorithm

The problem of quantifying the vulnerability of graphs has received much attention nowadays, especially in the field of computer or communication networks. In a communication network, the vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modeling a network, the average lower 2-domination number of a graph is a measure of the graph vulnerability and it is defined by γ 2 a v ( G ) = 1 | V ( G ) | ∑ v ∈ V ( G ) γ 2 v ( G ) , where the lower 2-domination number, denoted by γ 2 v ( G ) , of the graph G relative to v is the minimum cardinality of 2-domination set in G that contains the vertex v. In this paper, the average lower 2-domination number of wheels and some related networks namely gear graph, friendship graph, helm graph and sun flower graph are calculated. Then, we offer an algorithm for computing the 2-domination number and the average lower 2-domination number of any graph G

The average lower reinforcement number of a graph

Let G = (V(G),E(G)) be a simple undirected graph. The reinforcement number of a graph is a vulnerability parameter of a graph. We have investigated a refinement that involves the average lower reinforcement number of this parameter. The lower reinforcement number, denoted by re∗(G), is the minimum cardinality of reinforcement set in G that contains the edge e∗ of the complement graph G̅ . The average lower reinforcement number of G is defined by rav(G)=1/|E(G̅)| ∑e** ∈ E(G̅) re*(G) .In this paper, we define the average lower reinforcement number of a graph and we present the exact values for some well−known graph families

Combining the Concepts of Residual and Domination in Graphs

###EgeUn###Let G = (V (G), E(G)) be a simple undirected graph. The domination and average lower domination numbers are vulnerability parameters of a graph. We have investigated a refinement that involves the residual domination and average lower residual domination numbers of these parameters. The lower residual domination number, denoted by gamma(R)(uk)(G), is the minimum cardinality of dominating set in G that received from the graph G where the vertex v(k) and all links of the vertex v(k) are deleted. The residual domination number of graphs G is defined as gamma(R)(G) = minv(k)is an element of V(G){gamma(R)(vk)(G)} . The average lower residual domination number of G is de- fined by gamma(R)(av)(G) = 1/vertical bar V(G)vertical bar Sigma(vk is an element of V(G)) gamma(R)(vk)(G). In this paper, we define the residual domination and the average lower residual domination numbers of a graph and we present the exact values, upper and lower bounds for some graph families

Analysis of Vulnerability of Some Transformation Networks

In several different applications and contexts, networks are essential frameworks and appear.Vulnerability value is a measure of the network's durability in the face of damage that may lead to a reduction or complete loss of the network's particular functionality. The domination number and it's types can be used network vulnerability parameters. Recently, the disjunctive total domination number has been defined by Henning and Naicker. In this paper, the disjunctive total domination numbers of the transformation graph G(++z )when z = {+, -} of some graphs G have been obtained. Furthermore, some new general results have been given for the parameter mentioned above

On the bondage, strong and weak bondage numbers in Complementary Prism Graphs

Let G = (V (G), E(G)) be a simple undirected graph of order n, and let S subset of (G). If every vertex in V (G) - S is adjacent to at least one vertex in S, then the set S is called a dominating set. The domination number of G is the minimum cardinality taken over all sets of S, and it is denoted by gamma(G). Recently, the effect of one or more edges deletion on the domination number has been examined in many papers. Let F subset of E(G). The bondage number b(G) of G is the minimum cardinality taken over all sets of F such that gamma(G - F) > gamma(G). In the literature, a lot of domination and bondage parameters have been defined depending on different properties. In this paper, we investigate the bondage, strong and weak bondage numbers of complementary prism graphs of some well-known graph families

RELATIONSHIPS BETWEEN VERTEX ATTACK TOLERANCE AND OTHER VULNERABILITY PARAMETERS

WOS: 000405570600003Let G(V,E) be a simple undirected graph. Recently, the vertex attack tolerance (VAT) of G has been defined as ?(G) = min {|S| / |V-S-Cmax (G-S)|+1 : S ? V} , where Cmax(G - S) is the order of a largest connected component in G - S. This parameter has been used to measure the vulnerability of networks. The vertex attack tolerance is the only measure that fully captures both the major bottlenecks of a network and the resulting component size distribution upon targeted node attacks. In this article, the relationships between the vertex attack tolerance and some other vulnerability parameters, namely connectivity, toughness, integrity, scattering number, tenacity, binding number and rupture degree have been determined