Consider the curve C = {(t, f(t): 0 < t < 1}, where f is absolutely continuous on [0, 1]. Then C has finite length, and if Ue is the &neighborhood of f in the uniform norm, we compare the length of the shortest path in UE with the length of f. Our main result establishes necessary and sufficient conditions on f such that the difference of these quantities is of order e as e -* 0. We also includ...

Article history: Available online 10 April 2013

We examine the existence of tangent hyperplanes to subriemannian balls. Strictly abnormal shortest paths are allowed.

We consider two approaches that model timetable information in public transportation systems as shortestpath problems in weighted graphs. In the time-expanded approach every event at a station, e.g., the departure of a train, is modeled as a node in the graph, while in the time-dependent approach the graph contains only one node per station. Both approaches have been recently considered for the...

Valence-weightings are considered for shortest-path distance moments, as well as related weightings for the so-called “Wiener” polynomial. In the case of trees the valence-weighted quantities are found to be expressible as a combination of unweighted quantities. Further the weighted quantities for a so-called “thorny” graph are considered and shown to be related to the weighted and unweighted q...

Abstract. Let G = (N,A) be an undirected graph with n nodes and m arcs, a designated source node s and a sink node t. This paper addresses sensitivity analysis questions concerning the shortest s-t path (SP) problem in G and the maximum capacity s-t path (MCP) problem in G. Suppose that P* is a shortest s-t path in G with respect to a nonnegative distance vector c. For each arc e ∈ A, the lower...

We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the number and length of shortest paths between nodes. For our example classification problem, we consider the task of classifying random graphs from two well-known families, by the number of clusters they contain. We verify empirically that the ...

We introduce a path-based measure of convexity to be used in assessing the compactness of legislative districts. Our measure is the probability that a district will contain the shortest path between a randomly selected pair of its’ points. The measure is defined relative to exogenous political boundaries and population distributions. JEL classification: D72, K00, K19