Integrity and domination integrity of gear graphs

C.A. Barefoot, et. al. [4] introduced the concept of the integrity of a graph. It is an useful measure of vulnerability and it is defined as follows. I(G) = min{|S| + m(G − S) : S ⊂ V (G)}, where m(G − S) denotes the order of the largest component in G − S. Unlike the connectivity measures, integrity shows not only the difficulty to break down the network but also the damage that has been caused. A subset S of V (G) is said to be an I-set if I(G) = |S| + m(G − S). We introduced a new vulnerability parameter in [4],namely domination integrity of a graph G. It is a defined as DI(G) = min{|S| + m(G − S)}, where S is a dominating set of G and m(G − S) denotes the order of the largest component in G − S. K.S. Bagga,et. al. [2] gave a formula for I(K2 × Cn). In this paper, we give a correct formula for I(K2 × Cn). We find some results on the integrity and domination integrity of gear graphs.Publisher's Versio

Domination Integrity of Some Path Related Graphs

The stability of a communication network is one of the important parameters for network designers and users. A communication network can be considered to be highly vulnerable if the destruction of a few elements cause large damage and only few members are able to communicate. In a communication network several vulnerability measures like binding number, toughness, scattering number, integrity, tenacity, edge tenacity and rupture degree are used to determine the resistance of network to the disruption after the failure of certain nodes (vertices) or communication links (edges). Domination theory also provides a model to measure the vulnerability of a graph network. The domination integrity of a simple connected graph is one such measure. Here we determine the domination integrity of square graph of path as well as the graphs obtained by composition (lexicographic product) of two paths

Neighbor Isolated Tenacity of Graphs

The tenacity of a graph is a measure of the vulnerability of a graph. In this paper we investigate a refinement that involves the neighbor isolated version of this parameter. The neighbor isolated tenacity of a noncomplete connected graph G is defined to be NIT(G) = min {|X|+ c(G/X) / i(G/X), i(G/X) ≥ 1} where the minimum is taken over all X, the cut strategy of G , i(G/X)is the number of components which are isolated vertices of G/X and c(G/X) is the maximum order of the components of G/X. Next, the relations between neighbor isolated tenacity and other parameters are determined and the neighbor isolated tenacity of some special graphs are obtained. Moreover, some results about the neighbor isolated tenacity of graphs obtained by graph operations are given

Edge-Neighbor-Rupture Degree of Graphs

The edge-neighbor-rupture degree of a connected graph is defined to be , where is any edge-cut-strategy of , is the number of the components of , and is the maximum order of the components of . In this paper, the edge-neighbor-rupture degree of some graphs is obtained and the relations between edge-neighbor-rupture degree and other parameters are determined

A Vulnerability Measure of <i>k</i>-Uniform Linear Hypergraphs

Vulnerability refers to the ability of a network to continue functioning when part of the network is either naturally damaged or targeted for attack. In this paper, the rupture degree of graphs is employed to measure the vulnerability of uniform linear hypergraphs. First, we discuss the bounds of the rupture degrees of k-uniform linear hypergraphs. Then, we give a recursive algorithm for computing the rupture degree of k-uniform hypertrees

INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE

The rupture degree of a graph is a measure of the vulnerability of a graph. In this paper we investigate a refinement, that involves the weak version of the parameter. The weak rupture degree of a connected graph G is defined to be R-w(G) = max{w(G - S) - |S| - m(e)(G - S) : S subset of V (G), w (G - S) > 1} where w(G - S) is the number of the components of G - S and m(e)(G - S) is the number of edges of the largest component of G - S. Like the rupture degree itself, this is a measure of the vulnerability of a graph, but it is more sensitive. This paper, the weak-rupture degree of some special graphs are obtained and sonic bounds of the weak-rupture degree are given. Moreover some results about the weak-rupture degree of graphs obtained by graph operations are given