# Spanning Tree Auxiliary Graphs

## TL;DR

This paper introduces a new class of auxiliary graphs based on spanning trees, characterizes their structure, and develops algorithms to analyze their relationship with original graphs.

## Contribution

It provides a mathematical characterization of spanning tree auxiliary graphs and algorithms for computing preimages, advancing understanding of their structural properties.

## Key findings

- Characterization of graphs that are spanning tree auxiliary graphs.
- Algorithms for computing basic preimages of auxiliary graphs.
- Relations between parameters of auxiliary and original graphs.

## Abstract

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.