Optimally Reliable Graphs for Both Vertex and Edge Failures

Abstract

We consider networks in which both the nodes and the links may fail. We represent the network by an undirected graph G. Vertices of the graph fail with probability p and edges of the graph fail with probability q, where all failures are assumed independent. We shall be concerned with minimising the probability P(G) that G is disconnected for graphs with given numbers of vertices and edges. We show how to construct these optimal graphs in many cases when p and q are 'small'.

Local Optimality of Mixed Reliability for Several Classes of Networks with Fixed Sizes

“…) is said to be an i-j mixed cut set of G. Let M ij (G) denote the number of i-j mixed cut sets in a graph G. It is shown in the Ref. [20] that, in the vertex-and-edge fault model, the disconnected probability of G, denoted by P(G; p, q), can be expressed as…”

“…For the vertex-and-edge fault model, Evans and Smith [19] made a connection between the following two cases: small p, q = 0 and small q, p = 0. Smith [20] showed that when p and q are small, a small-mixed-cut-set-optimal graph has the minimum unconnected probability among the graphs with given numbers of vertices and edges, and gave the method of how to construct these optimal graphs in many cases. Chen and He [21] in 2004 gave the upper and lower bounds of the mixed reliability of networks with unreliable vertices and edges.…”