A partition M is k-optimal if M minimizes BdM). For instance, if G has a Hamiltonian path J.Lo, then M = W.o} is a k -optimal partition. So the k -optimal partitions extend in some sense the concept of Hamiltonian paths. The theorem of Greene and Kleitman [10], which extends the Dilworth theorem [5], shows an important property of k-optimal partitions for the graph of a partially ordered set (i...

Let G be an Eulerian directed graph with an arc-labeling such that arcs going out from the same vertex have di erent labels. In this work, we present an algorithm to construct the Eulerian trail starting at an arbitrary vertex v of minimum lexicographical label among labels of all Eulerian trails starting at this vertex. We also show an application of this algorithm to construct the minimal de ...

We investigate the structure of global inter-firm relationships using a unique dataset containing information on customers, suppliers, licensors, licensees and strategic alliances for each of 412,814 major incorporated non-financial firms in the world. We focus on three different networks: customer-supplier network, licensee-licensor network, and strategic alliance network. In/out-degree distri...

A digraph D is k-transitive if the existence of a directed path (v0, v1, . . . , vk), of length k implies that (v0, vk) ∈ A(D). Clearly, a 2-transitive digraph is a transitive digraph in the usual sense. Transitive digraphs have been characterized as compositions of complete digraphs on an acyclic transitive digraph. Also, strong 3 and 4-transitive digraphs have been characterized. In this work...

We consider a class of Boolean networks called and-nets, and we address the question of whether the absence of negative cycle in local interaction graphs implies the existence of a fixed point. By defining correspondences with the notion of kernel in directed graphs, we prove a particular case of this question, and at the same time, we prove new theorems in kernel theory, on the existence and u...

In graph theory, Menger’s Theorem is the basis of many results concerning notions of connectedness. There are many versions, in this paper the following will be useful. Let G = (V, E) be a directed graph and A, B disjoint subsets of V with no directed edge from A to B. A subset of V -A U B is called an (A, B) separating vertex-set if every directed path from A to B intersects F. A set of direct...

This paper presents new algorithms for computing single source shortest paths (SSSPs) in a nearly acyclic directed graph G. The first part introduces higher-order decomposition. This decomposition is an extension of the technique of strongly connected component (sc-component) decomposition. The second part presents a new method for measuring acyclicity based on modifications to two existing met...

We consider directed polymers in random environment. Under mild assumptions on the environment, we prove here: (i) equivalence of decay rate of the partition function with some natural localization properties of the path, (ii) quantitative estimates of the decay of the partition function in dimensions one or two, or at sufficiently low temperature, (iii) existence of quenched free energy. In pa...

In a graph G, a spanning tree T is called a tree t-spanner of G if the distance between any two vertices in T is at most t times their distance in G. While the complexity of finding a tree t-spanner of a given graph is known for any fixed t Þ 3, the case t 5 3 still remains open. In this article, we show that each directed path graph G has a tree 3-spanner T by means of a linear-time algorithm ...

This thesis introduces a new acceleration heuristic for shortest path queries, called the PCD algorithm (Precomputed Cluster Distances). PCD precomputes shortest path distances between the partitions of the input graph, which can be obtained by any graph partitioning method. Since the number of partitions can be varied between one and the number of nodes, the method presents an interpolation be...