Suppose that every edge of a graph G (finite and undirected) is independently operational with probability p ∈ [0, 1]. The all terminal reliability of G is the probability that all vertices can communicate. It was conjectured that among all graphs with n vertices and m edges there always exists a most optimal graph, that is, one whose all terminal reliability is at least as large as any other such graph, no matter what the value of p. For each n ≥ 6, a single value of m was found for which the restriction of the conjecture to simple graphs failed, but it remained open as to whether most optimal graphs exist when multiple edges are allowed. We show that in fact for a given n ≥ 6, there are several values of m for which a most optimal simple graph does not exist. Moreover, we prove that including multiple edges still does not introduce a most optimal graph, disproving for the first time the conjecture for general graphs. In contrast, it will be shown that for a given n and m, there always exists a least optimal graph.

A variety of probabilistic notions of network reliability of graphs and digraphs have been proposed and studied since the early 1950s. Although grounded in the engineering and logistics of network design and analysis, the research also spans pure and applied mathematics, with connections to areas as diverse as combinatorics and graph theory, combinatorial enumeration, optimization, probability theory, real and complex analysis, algebraic topology, commutative algebra, the design and analysis of algorithms, and computational complexity. In this paper we describe the landscape of various notions of network reliability, the roads well traveled, and some that appear likely to lead to meaningful and important journeys.

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We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in [0, 1/2].