On the ensemble of optimal identifying codes in a twin-free graph

Let G = (V, E) be a graph. For v is an element of V and r >= 1, we denote by B-G,B- r (v) the ball of radius r and centre v. A set C subset of V is said to be an r-identifying code if the sets B-G,B- r (v) boolean AND C, v is an element of V, are all nonempty and distinct. A graph G which admits an r-identifying code is called r-twin-free, and in this case the smallest size of an r-identifying code is denoted by gamma(r)(G). We study the ensemble of all the different optimal r-identifying codes C, i.e., such that broken vertical bar C broken vertical bar = gamma(r)(G). We show that, given any collection A of k-subsets of V-1 = {1, 2, . . . , n}, there is a positive integer m, a graph G = (V, E) with V = V-1 boolean OR V-2, where V-2 = {n + 1, . . . , n + m}, and a set S subset of V-2 such that C subset of V is an optimal r-identifying code in G if, and only if, C = A boolean OR S for some A is an element of A. This result gives a direct connection with induced subgraphs of Johnson graphs, which are graphs with vertex set a collection of k-subsets of V1, with edges between any two vertices sharing k - 1 elements

Can strategizing in round-robin subtournaments be avoided?

This paper develops a mathematical model of strategic manipulation in complex sports competition formats such as the soccer world cup or the Olympic games. Strategic manipulation refers here to the possibility that a team may lose a match on purpose in order to increase its prospects of winning the competition. In particular, the paper looks at round-robin tournaments where both first- and second-ranked players proceed to the next round. This standard format used in many sports gives rise to the possibility of strategic manipulation, as exhibited recently in the 2012 Olympic games. An impossibility theorem is proved which demonstrates that under a number of reasonable side-constraints, strategy-proofness is impossible to obtain

Robust Bounds on Choosing from Large Tournaments

Tournament solutions provide methods for selecting the "best" alternatives from a tournament and have found applications in a wide range of areas. Previous work has shown that several well-known tournament solutions almost never rule out any alternative in large random tournaments. Nevertheless, all analytical results thus far have assumed a rigid probabilistic model, in which either a tournament is chosen uniformly at random, or there is a linear order of alternatives and the orientation of all edges in the tournament is chosen with the same probabilities according to the linear order. In this work, we consider a significantly more general model where the orientation of different edges can be chosen with different probabilities. We show that a number of common tournament solutions, including the top cycle and the uncovered set, are still unlikely to rule out any alternative under this model. This corresponds to natural graph-theoretic conditions such as irreducibility of the tournament. In addition, we provide tight asymptotic bounds on the boundary of the probability range for which the tournament solutions select all alternatives with high probability.Comment: Appears in the 14th Conference on Web and Internet Economics (WINE), 201

Resistance to autosomal dominant Alzheimer's disease in an APOE3 Christchurch homozygote: a case report.

We identified a PSEN1 (presenilin 1) mutation carrier from the world's largest autosomal dominant Alzheimer's disease kindred, who did not develop mild cognitive impairment until her seventies, three decades after the expected age of clinical onset. The individual had two copies of the APOE3 Christchurch (R136S) mutation, unusually high brain amyloid levels and limited tau and neurodegenerative measurements. Our findings have implications for the role of APOE in the pathogenesis, treatment and prevention of Alzheimer's disease