Counting the Number of Minimal Paths in Weighted Coloured-Edge Graphs

TL;DR

This paper establishes tight bounds on the number of minimal paths in weighted coloured-edge graphs, which model multi-modal networks, providing insights into their complexity and expected path counts.

Contribution

It introduces a tight upper bound on the minimal path set size and an expected value bound for weighted bicoloured-edge graphs.

Findings

Tight upper bound on minimal path set cardinality.

Bound on expected number of minimal paths in bicoloured-edge graphs.

Enhanced understanding of path complexity in multi-modal networks.

Abstract

A weighted coloured-edge graph is a graph for which each edge is assigned both a positive weight and a discrete colour, and can be used to model transportation and computer networks in which there are multiple transportation modes. In such a graph paths are compared by their total weight in each colour, resulting in a Pareto set of minimal paths from one vertex to another. This paper will give a tight upper bound on the cardinality of a minimal set of paths for any weighted coloured-edge graph. Additionally, a bound is presented on the expected number of minimal paths in weighted bicoloured-edge graphs.