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On Isolated Rupture Degree of Graphs
Abstract
The isolated rupture degree for a connected graph G is defined as ir(G) = max{i(G - S) - vertical bar S vertical bar - m(G - S) : S (sic) C(G)}, where i(G - S) and m(G - S), respectively, denote the number of components which are isolated vertices and the order of a largest component in G - S. C(G) denotes the set of all cut-sets of G. The isolated rupture degree is a new graph parameter which can be used to measure the vulnerability of networks. In this paper, we give isolated rupture degrees of several specific classes of graphs. Formulas for the isolated rupture degree of join graphs and some bounds of the rupture degree are given. We also determine the isolated rupture degree of grids, and that of the hypercubes as a special case.
... A communication network is modelled as an undirected and unweighted graph, where a processor (station) is represented as a vertex and a communication link between processors (stations) as an edge between corresponding vertices. When we use a graph to model a network, based on the above three quantities, a number of graph parameters have been proposed for measuring the vulnerability of networks, such as connectivity, toughness [6], scattering number [11], integrity [1], tenacity [7], rupture degree [19], isolated rupture degree [16,17] and their edge-analogues. ...
... Hence, the isolated scattering number is a reasonable parameter for distinguishing the vulnerability of these graphs. The less the isolated scattering number of a network the more stable it is considered to be [16]. Wang et al. [21] gave formulas for the isolated scattering number of join graphs and some bounds of the isolated scattering number, and they also give a recursive algorithm for computing the isolated rupture degree of trees. ...
... There are many well known measures of vulnerability based on subsets of vertices. Some are more usual as connectivity, domination number, domatic number and independence number; other ones are more recent as toughness [1], binding number [2], scattering number [3], integrity [4], tenacity [5], rupture degree [6] and some variations as domination integrity [7] and isolated rupture degree [8] (as cited in [9]). Relationships between these invariants were studied in [4,10]. ...
The scattering number of a graph G was defined by Jung in 1978 as sc(G)=max{ω(G-S)-|S|,S⊆V,ω(G-S)≠1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$sc(G) = max \{ \omega (G - S) - |S|, S \subseteq V, \omega (G - S) \ne 1\}$$\end{document} where ω(G-S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (G - S) $$\end{document} is the number of connected components of the graph G-S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G-S$$\end{document}. It is a measure of vulnerability of a graph and it has a direct relationship with its toughness. Strictly chordal graphs, also known as block duplicate graphs, are a subclass of chordal graphs that includes block and 3-leaf power graphs. In this paper we present a linear time solution for the determination of the scattering number and scattering set of strictly chordal graphs. We show that, although the knowledge of the toughness of the strictly chordal graphs is helpful, it is far from sufficient to provide an immediate result for determining the scattering number.
Computer or communication networks are so designed that they do not easily get disrupted under external attack. Moreover, they are easily reconstructed when they do get disrupted. These desirable properties of networks can be measured by various parameters, such as connectivity, toughness and scattering number. Among these parameters, the isolated scattering number is a comparatively better parameter to measure the vulnerability of networks. In this paper we first prove that for split graphs, this number can be computed in polynomial time. Then we determine the isolated scattering number of the Cartesian product and the Kronecker product of special graphs and special permutation graphs.
We introduce a new graph parameter, the rupture degree. The rupture degree for a complete graph Kn is defined as 1−n, and the rupture degree for an incomplete connected graph G is defined by r(G)=max{ω(G−X)−|X|−m(G−X):X⊂V(G), ω(G−X)>1}, where ω(G−X) is the number of components of G−X and m(G−X) is the order of a largest component of G−X. It is shown that this parameter can be used to measure the vulnerability of networks. Rupture degrees of several specific classes of graphs are determined. Formulas for the rupture degree of join graphs and some bounds of the rupture degree are given. We also obtain some Nordhaus–Gaddum type results for the rupture degree.
The rupture degree of a noncomplete connected graph G is defined by r(G) = max{ω(G - X) - |X| - m(G - X) : X ⊂ V(G), ω(G - X) ≥ 2}, where ω(G - X) denotes the number of components in the graph G - X. For a complete graph K<sub>n</sub>, we define r(K<sub>n</sub>) = 1 - n. This parameter can be used to measure the vulnerability of a graph. To some extent, it represents a trade-off between the amount of work done to damage the network and how badly the network is damaged. In this paper, we prove that the problem of computing the rupture degree of a graph is NP-complete. We obtain the rupture degree of the Cartesian product of some special graphs and also give the exact values or bounds for the rupture degrees of Harary graphs.
The concept of an order-theoretical tree is generalized to the notion of a multitree. The comparability graphs of multitrees are characterized and studied with respect to minimal path coverings.