No abstract

We consider a particular case of the Fleet Quickest Routing Problem (FQRP) on a grid graph of m 9 n nodes that are placed in m levels and n columns. Starting nodes are placed at the first (bottom) level, and nodes of arrival are placed at the mth level. A feasible solution of FQRP consists in n Manhattan paths, one for each vehicle, such that capacity constraints are respected. We establish m*, i.e. the number of levels that ensures the existence of a solution to FQRP in any possible permutation of n destinations. In particular, m* is the minimum number of levels sufficient to solve any instance of FQRP involving n vehicles, when they move in the ways that the literature has until now assumed. Existing algorithms give solutions that require, for some values of n, more levels than m*. For this reason, we provide algorithm CaR, which gives a solution in a graph m* 9 n, as a minor contribution.

No abstract

The k-centrum of a graph G = ( V , E ) is the set of vertices each minimizing the sum of the distances from it to the corresponding k farthest vertices of G. In this article properties of k-centrum of trees, where edges have positive real lengths, are investigated.

No abstract