On the construction of optimally reliable graphs

Abstract

We consider the reliability of regular graphs of degree k in which the vertices of the graph fail independently with probability p. The probability of disconnection is minimized for small values of p if the graph has connectivity k and has the smallest number of vertex cut sets with k vertices. In this paper, we show how to construct such graphs.

Optimally Reliable Graphs for Both Vertex and Edge Failures

“…This statement will be made exact in Theorem 3.3. Families of regular graphs with K = k and M k0 minimised have been constructed by Smith [12] and Smith and Doty [13]. In the following section we prove a general result, which can be used to verify that many of these graphs are small-mixed-cut-set-optimal.…”

“…The families of graphs of degree 3 with N > 8 and degree 4 in [12] and the regular graphs of [13] all satisfy the girth condition of Theorem 3.1, so by Theorem 3.2 are small-mixed-cut-set-optimal and Theorem 3.3 applies. In Figure 5 we give two typical examples of the regular graphs constructed in [13], by displaying their quotient graphs. Other constructions of [12] do not satisfy the required girth condition, and the values of M uk _j for j = 0,1,2,..., k -1 require direct determination.…”

“…Before proving our main result, we give some examples that show that this is not always the case and also that the girth condition we shall use cannot be removed. Some of these examples are shown by displaying their quotient graphs [12], [13]. , so the graph G 2 is more reliable than G, for small />,?…”

“…The regular graphs of [12], [13] have degree k, vertex connectivity k, no vertex cut sets with k vertices that are not neighbour sets of a vertex and M k 0 minimal. The families of graphs of degree 3 with N > 8 and degree 4 in [12] and the regular graphs of [13] all satisfy the girth condition of Theorem 3.1, so by Theorem 3.2 are small-mixed-cut-set-optimal and Theorem 3.3 applies. In Figure 5 we give two typical examples of the regular graphs constructed in [13], by displaying their quotient graphs.…”

“…We shall be concerned with minimising the probability P(G) that G is disconnected for graphs with given numbers of vertices and edges. This problem has normally been dealt with in cases when either p = 0 ([1], [2], [3] and [8]) or q = 0 ( [6], [9], [10], [11], [12] and [13]). In section 2 we introduce the definition of small-mixed-cut-set-optimal graphs.…”