On reliability of graphs with node failures

We consider the reliability of graphs for which nodes fail independently of each other with a constant probability 1 p. The reliability of a graph is defined to be the probability that the induced subgraph of surviving nodes is connected. A graph is said to be uniformly best when, for all choices of p, it is most reliable in the class of graphs with the same number of nodes and same number of edges. In this paper, we first extend the existing known set of uniformly best graphs. Next, we show that most classes of sparse graphs do not contain a uniformly best graph. Finally, we introduce the important notions of locally best and asymptotically best graphs and illustrate these concepts with a detailed study of graphs having the same number of nodes and edges.