  In this paper, we define a class of auxiliary graphs associated with simple
undirected graphs. This class of auxiliary graphs is based on the set of
spanning trees of the original graph and the edges constituting those spanning
trees. A class of auxiliary graphs can be viewed as a function from the class
of graphs to the class of graphs. We provide mathematical characterisation of
graphs which are the spanning tree auxiliary graphs of some simple graph. Since
the class of spanning tree auxiliary graphs of graphs do not have unique
preimages (the forward function is not injective), we derive precisely the
classes of graphs which have the same auxiliary graph. We design algorithms for
computing a basic preimage and define rules to get other solutions for the same
auxiliary graph. We also obtain several results expressing parameters of the
auxiliary graph in terms of (not necessarily the same) parameters of the
original graph.

Combinatorics