  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.
