Minimum Labelling bi-Connectivity

TL;DR

This paper investigates the problem of finding the minimum set of labels in a graph that ensures bi-connectivity, extending the well-known minimum labelling spanning tree problem to a more complex connectivity requirement.

Contribution

It introduces the minimum labelling bi-connectivity problem, considering both edge and vertex bi-connectivity, and discusses initial solution approaches for this NP-hard problem.

Findings

Preliminary investigation into the problem.

Development of solution approaches.

Extension of labelling problems to bi-connectivity.

Abstract

A labelled, undirected graph is a graph whose edges have assigned labels, from a specific set. Given a labelled, undirected graph, the well-known minimum labelling spanning tree problem is aimed at finding the spanning tree of the graph with the minimum set of labels. This combinatorial problem, which is NP-hard, can be also formulated as to give the minimum number of labels that provide single connectivity among all the vertices of the graph. Here we consider instead the problem of finding the minimum set of labels that provide bi-connectivity among all the vertices of the graph. A graph is bi-connected if there are at least two disjoint paths joining every pair of vertices. We consider both bi-connectivity concept: the edge bi-connectivity where these paths cannot have a common edge and the vertex bi-connectivity where the paths cannot have a common vertex. We describe our preliminary investigation on the problem and provide the details on the solution approaches for the problem under current development.