By a new type of finite size scaling analysis on the square lattice, and by renormalization group calculations on hierarchical lattices we investigate the effects of dilution on optimal undirected self-avoiding paths in a random environment. The behaviour of the optimal paths remains the same as for directed paths in undiluted medium, as long as forbidden bonds are not exceeding the percolation...

An arborescence in a digraph is a tree directed away from its root. A classical theorem of Edmonds characterizes which digraphs have λ arc-disjoint arborescences rooted at r. A similar theorem of Menger guarantees λ strongly arc disjoint rv-paths for every vertex v, where “strongly” means no two paths contain a pair of symmetric arcs. We prove that if a directed graph D contains two arc-disjoin...

We prove the NP-completeness of the integer multiflow problem in planar graphs, with the following restrictions: there are only two demand edges, both lying on the infinite face of the routing graph. This was one of the open challenges concerning disjoint paths, explicitly asked by Müller [5]. It also strengthens Schwärzler’s recent proof of one of the open problems of Schrijver’s book [9], abo...

We combine the well known Shortest Paths (SP) problem and the Bottleneck Paths (BP) problem to introduce a new problem called the Shortest Paths for All Flows (SP-AF) problem that has relevance in real life applications. We first solve the Single Source Shortest Paths for All Flows (SSSP-AF) problem on directed graphs with unit edge costs in O(mn) worst case time bound. We then present two algo...

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger’s theorem gives a generalization of the theorem of Tverberg on non-separated partitions.

Across languages and event types (agentive and non-agentive motion, transfer, change of state, attach/detach), goal paths are privileged over source paths in the linguistic encoding of events. Furthermore, some linguistic analyses suggest that goal paths are more central than source paths in the semantic and syntactic structure of motion verbs. However, in the non-linguistic memory of children ...