Abstract. We construct an infinite binary word with critical exponent 3 that avoids abelian 4-powers. Our method gives an algorithm to determine whether certain types of morphic sequences avoid additive powers. We also show that there are [Formula: see text] binary words of length [Formula: see text] that avoid abelian 4-powers, which improves on previous estimates.

Abstract 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.

Given a graph G whose edges are perfectly reliable and whose nodes each operate independently with probability the node reliability of G is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce; it is the analogous node measure of robustness to the well studied all‐terminal reliability, where the nodes are perfectly reliable but the edges fail randomly. In sharp contrast to what is known about the roots of the all‐terminal reliability polynomial, we show that the node reliability polynomial of any connected graph on at least three nodes has a nonreal polynomial root, the collection of real roots of all node reliability polynomials is unbounded, and the collection of complex roots of all node reliability polynomials is dense in the entire complex plane. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(3), 238–246 2016