Let −→ Λ be an infinite, locally finite oriented multi-graph with C− → Λ finite and strongly connected, and let p < pH( −→ Λ ). Menshikov’s theorem states that there exists an α > 0 s.t. Pp(x n →) ≤ exp(−αn/(logn)) for all sites x and n ≥ 2. In this paper, we begin backwards, explaining the importance of this theorem and then giving a sketch of the proof, emphasizing the intuition and motivatio...

Given a digraph D, consider the model where each vertex is always operational, but the edges are independently operational with probability p. The strongly connected reliability of D, scRel(D, p), is the probability that the spanning subgraph of D consisting of the operational edges is strongly connected. One can view strongly connected reliability as the probability that any vertex can send in...

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore-like bound in terms of its diameter k and the maximum out-degrees (d1; d2) of its partite sets of vertices. It has been proved that, when d1d2 > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist w...

let $a$ be a nite dimensional $k-$algebra and $r$ be a locally bounded category such that $r rightarrow r/g = a$ is a galois covering de ned by the action of a torsion-free group of automorphisms of $r$. following [30], we provide criteria on the convex subcategories of a strongly simply connected category r in order to be a cycle- nite category and describe the module category of $a$. we pro...

Parity games have important practical applications in formal verification and synthesis, especially to solve the model-checking problem of the modal mu-calculus. They are also interesting from the theory perspective, because they are widely believed to admit a polynomial solution, but so far no such algorithm is known. We propose a new algorithm to solve parity games based on learning tangles, ...

This technical communique represents a generalization of the convergence analysis for the consensus algorithm proposed in Priolo et al. (2014). Although the consensus was reached for any strongly connected weighted digraphs (SCWD), the convergence analysis provided in Priolo et al. (2014) was only valid for diagonalizable matrices encoding a SCWD. The result we present here generalizes the prev...

The AllDifferent constraint is a crucial component of any constraint toolkit, language or solver, since it is very widely used in a variety of constraint models. The literature contains many different versions of this constraint, which trade strength of inference against computational cost. In this paper, we focus on the highest strength of inference (enforcing a property known as generalised a...

Given a set V of n sensor node distributed on a 2-dimensional plane and a source node s ∈ V, the interference problem deals with assigning transmission range to each v ∈ V such that the members in V maintain connectivity predicate P , and the maximum/total interference is minimum. We propose algorithm for both minimizing maximum interference andminimizing total interference of the networks. For...

The oriented diameter of a bridgeless connected undirected (bcu ) graph G is the smallest diameter among all the diameters of strongly connected orientations of G. We study algorithmic aspects of determining the oriented diameter of a chordal graph. We (a) construct a linear-time approximation algorithm that, for a given chordal bcu graph G, finds a strongly connected orientation of G with diam...

Let G be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of G that maintains respectively: the 2-edge-connected blocks of G (2EC-B); the 2-edge-connected components of G (2EC-C); both the 2-edge-connected blocks and the 2edge-connected components of G (2EC-B-C). All three problems are NPhar...