Abstract We describe a polynomial‐time algorithm for deciding whether a given distance graph with a finite number of vertices is connected. This problem was conjectured to be NP‐hard in Draque Penso et al. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph's node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece of information, we can compute the graph's diameter and average minimum distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded monomial ordering. This result permits the use of Gröbner bases for computing the ideal and finding an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter d, all of them associated to irreducible monomial ideals.

Let p be a prime and let c be an integer modulo p. The Pollard generator is a sequence (un) of pseudorandom numbers defined by the relation un+1equivun2+c mod p. It is shown that if c and 9/14 of the most significant bits of two consecutive values un,un+1of the Pollard generator are given, one can recover in polynomial time the initial value u0with a probabilistic algorithm. This result is an improvement of a theorem in a recent paper which requires that 2/3 of the most significant bits be known