C. Suffel

• Stevens Institute of Technology

Consider a graph where edges can fail but nodes do not. However, when an edge fails, its endnodes are subverted, i.e., removed from the graph. Given a threshold value k >= 1, the surviving subgraph produced by the failure of edges and the subversion of the endnodes of those edges is said to be in a failure state if all of its components have order...

If a spy network is modeled as a graph in which the nodes represent the spies and the edges are the communication links between spies, then consider the scenario where the interception of a link gives up both endnode spies. In graph-theoretic terms, edges fail and nodes do not, but when they do, the endnodes are subverted, i.e., they are removed. G...

A corollary of the Kirchhoff matrix-tree theorem is used to find the number of spanning trees of a graph via the roots of the characteristic polynomial of the associated Laplacian matrix. Threshold graphs play a role in bounding the number of spanning trees from below, given that the number of nodes and edges are held fixed. Although other authors...

This book is concerned with the optimization problem of maximizing the number of spanning trees of a multigraph. Since a spanning tree is a minimally connected subgraph, graphs and multigraphs having more of these are, in some sense, immune to disconnection by edge failure. We employ a matrix-theoretic approach to the calculation of the number of s...

The traditional parameter used as a measure of vulnerability of a network modeled by a graph with perfect nodes and edges that may fail is edge connectivity λ. For the complete bipartite graph Kp,q where 1≤ p≤q, λ(Kp,q) = p. In this case, failure of the network means that the surviving subgraph becomes disconnected upon the failure of individual ed...

Let G be a graph with n nodes and e edges, where the nodes are perfectly reliable and the edges fail independently with equal probability ρ. A failure state exists if the surviving edges induce a graph having all components of order less than a preassigned threshold value k. The unreliability of G, Uk(G; ρ), is the probability of a failure state an...

The traditional vulnerability parameter connectivity is the minimum number of nodes needed to be removed to disconnect a network. Likewise, edge connectivity is the minimum number of edges needed to be removed to disconnect. A disconnected network may still be viable if it contains a sufficiently large component. Component order connectivity and co...

Given a graph G on n nodes and e edges, the component order connectivity parameter, κ c (k) (G) or simply κ c (k) , is defined as the minimum number of nodes that must be removed from a graph G in order to ensure that all remaining components have order less than some given threshold value, k. With neighbor-connectivity, the failure of a node cause...

Let G be a graph with n nodes and e edges and suppose each node is assigned a positive integer weight. Set W to be the sum of the weights of the nodes of G and w ¯ to be the maximum weight assigned to any node. Given k such that w ¯<k≤W, the weighted component edge connectivity of a graph is the minimum number of edges that must be removed from the...

In 1966, Chartrand proved that if the minimum degree of a graph is at least the floor of half the number of nodes, then its edge-connectivity equals its minimum degree. A more discriminating notion of edge-connectivity is introduced, called the k-component order edge-connectivity, which is the minimum number of edges required to be removed so that...

Assume that each vertex of a graph G is the possible location for an “intruder” such as a thief, or a saboteur, a fire in a facility or some possible processor fault in a computer network. A device at a vertex v is assumed to ...

The purpose of this article is to introduce several results concerning the analysis and synthesis of reliable or invulnerable networks. First, the notion of signed reliability domination of systems is described and some applications to reliability analysis are reviewed. Then the analysis problem is considered and a brief summary of the difficulty o...

There are networks that can be modeled by simple graphs, where edges are perfectly reliable but nodes are subject to failure, e.g. hardwired computer systems. One measure of the “vulnerability” of the network is the connectivity κ of the graph. Another, somewhat related, vulnerability parameter is the component order connectivity κ c (k) , i.e. the...

The component order edge connectivity parameter, λ c (k) (G) or simply λ c (k) is defined as the minimum number of edges that must be deleted from a graph G so that all components of the resulting subgraph have order less than k, where k is a predetermined threshold value. Formulas for λ c (k) have been derived for stars, paths, cycles, and complet...

It is well known that certain graph-theoretic extremal questions play a central role in the study of information network vulnerability. These extremal problems are special cases of the general question of realizability of graph invariants. For example a (p, Δ, δ, λ) graph is a graph having p-points, maximum degree Δ, minimum degree δ, and line-conn...

It is shown that a connected graph G spans an eulerian graph if and only if G is not spanned by an odd complete bigraph K(2m + 1, 2n + 1). A disconnected graph spans an eulerian graph if and only if it is not the union of the trivial graph with a complete graph of odd order. Exact formulas are obtained for the number of lines which must be added to...

Two classical parameters of graph connectivity are κ and λ, which are the minimum number of nodes and edges, respectively, that must be removed in order to disconnect the graph. Previously a new node parameter was introduced, κ c (k) , which is the minimum number of nodes that must be removed such that the remaining subgraph has no component of ord...

If a network is modeled by a simple graph in which the nodes represent the components of the network, then the connectivity of the graph is one measure of the “vulnerability” of the network. The authors have studied a somewhat related vulnerability parameter: the component order connectivity, i.e. the smallest number of nodes that must fail in orde...

A graph G with n nodes and e edges is said to be t-optimal if G has the maximum number of spanning trees among all graphs with the same number of nodes and edges as G. Hitherto, t-optimal graphs have been characterized for the following cases: 1.(a) n = sp, and e = (s(s - 1)/2)p2, when s and p are positive integers, and s > 1;2.(b) e ⩽ n + 2;3.(c)...

Most of the early work concerning “reliability” of a network assumed that the network was represented by a graph whose nodes were perfectly reliable but whose edges operated independently, with some probability p. The reliability of the network was defined to be the probability that the operating edges formed a connected spanning subgraph. With the...

Graph G has perfectly reliable nodes and edges that are subject to stochastic failure. The network reliability R is the probability that the surviving edges induce a spanning connected subgraph of G. Analysis problems concern determining efficient algorithms to calculate R, which is known to be NP-hard for general graphs. Synthesis problems concern...

The methods of network analysis grew out of a need to optimize performance, maximize reliability or minimize vulnerability, and/or reduce costs of large-scale systems involving flow of some kind, such as, communication networks, computer networks, transportation networks, and energy distribution networks. Of course graph theory pervades all the mod...

We introduce a new model for node failures which addresses the two faults of the usual residual node connectivity model.

We consider a probabilistic network in which the edges are perfectly reliable but the nodes fail with some known probabilities. The network is in an operational state if the surviving nodes induce a connected graph. The residual node connectedness reliability R(G) of a network G is the probability that the graph induced by the surviving nodes is co...

Motivated by connections with problems in network reliability, we explore the relationship between vertex neighborhoods Nu in a graph, focusing on neighborhood equality and inclusion (suitably modified for adjacent vertices). We survey recent work which shows that solutions to certain reliability extremal problems must be graphs in which the neighb...

This paper considers a probabilistic graph in which the points are perfectly reliable but the edges operate independently of one another, all with some known probability p. The graph G is in an operating state if the surviving edges induce a spanning connected subgraph of G. The all-terminal reliability R(G,p) of G is the probability that G is in a...

A well-known model for network reliability studies consists of an undirected graph with perfectly reliable nodes and equal and independent edge failure probabilities. The measure of reliability is then defined to be the probability that the graph is connected. A well-defined synthesis problem is to find the graph that minimizes the failure probabil...

This paper considers a probabilistic network in which the edges are perfectly reliable but the nodes fail with some known probabilities. The network is in an operational state if the surviving nodes induce a connected graph. The residual node connectedness reliability R(G) of a network G is the probability that the graph induced by the surviving no...

A well-known model in communication network reliability consists of an undirected graph G whose edges operate independently with the same probability p . Then the reliability, R ( G , p ) of G , is the probability that G is connected. It is known that R ( G , p ) is a polynomial in p and its coefficient of the least-order term is the number of span...

An important problem in reliability theory is to determine the reliability of a system from the reliability of its components. If E is a finite set of components, then certain subsets of E are prescribed to be the operating states of the system. A formation is any collection F of minimal operating states whose union is E. Reliability domination is...

A description is given of a way to obtain bounds which will be tight over the class of networks for all possible values of the edge failure rate. These bounds have been shown to be valid for a certain range of edge failure rates. The author discusses the known bounding methods before proposing the new approach

An undirected connected graph having failure probabilities associated with each edge is a classic model for network reliability studies. The network reliability is defined as the probability that the graph remains connected despite edge failures. It is known that the problem of calculating the network reliability is NP -hard, even when the edge fai...

Shannon's Sampling Theorem states that if the sampling frequency of a band limited function is chosen to be larger than twice the cutoff frequency of the signal then the cardinal series obtained from the samples coincides with the original function. Hence, we consider signals that are not band limited. In this case there will be an error between th...

Shannon's sampling theorem states that if the sampling frequency of a band-limited function is chosen to be larger than twice the cutoff frequency of the signal, then the cardinal series obtained from the samples coincides with the original function. Hence, signals that are not band limited are considered. In this case there will be an error betwee...

This paper presents some results regarding the design of reliable networks. The problem under consideration involves networks which are undirected graphs having equal and independent edge failure probabilities. The index of reliability is the probability that the network fails (becomes disconnected). For “small” edge failure probabilities and given...

Graphs with edge weights representing failure probabilities are models for network reliability problems. The network reliability is defined as the probability that the graph remains connected despite edge failures. Here we survey the known results which correspond to the synthesis problem of finding graphs which maximize the network reliability for...

In a recent paper, we gave a generalization of extremal problems involving certain graph-theoretic invariants. In that work, we defined a (p, Δ, δ, λ) graph as a graph having p points, maximum degree Δ, minimum degree δ, and line-connectivity λ. An arbitrary quadruple of integers (a, b, c, d) was called (p, Δ, δ, λ) realizable if there is a (p, Δ,...