On the Structure of the Graph of Unique Symmetric Base Exchanges of Bispanning Graphs

TL;DR

This paper investigates the structure of the graph of unique symmetric base exchanges in bispanning graphs, providing a classification and composition methods, and exploring their connectivity properties.

Contribution

It introduces a classification of bispanning graphs based on their subgraph and connectivity properties, and presents new composition methods for their unique exchange graphs.

Findings

Unique exchange graphs are Cartesian products for graphs with non-trivial bispanning subgraphs.

Bispanning graphs with vertex-connectivity two are 2-clique sums, and their exchange graphs can be combined accordingly.

A composition method at degree-three vertices constructs exchange graphs from three reduced bispanning graphs.

Abstract

Bispanning graphs are undirected graphs with an edge set that can be decomposed into two disjoint spanning trees. The operation of symmetrically swapping two edges between the trees, such that the result is a different pair of disjoint spanning trees, is called an edge exchange or a symmetric base exchange. The graph of symmetric base exchanges of a bispanning graph contains a vertex for every valid pair of disjoint spanning trees, and edges between them to represent all possible edge exchanges. We are interested in a restriction of these graphs to only unique symmetric base exchanges, which are edge exchanges wherein selecting one edge leaves only one choice for selecting the other. In this thesis, we discuss the structure of the graph of unique symmetric edge exchanges, and the open question whether these are connected for all bispanning graphs. Our composition method classifies bispanning graphs by whether they contain a non-trivial bispanning subgraph, and by vertex- and edge-connectivity. For bispanning graphs containing a non-trivial bispanning subgraph, we prove that the unique exchange graph is the Cartesian graph product of two smaller exchange graphs. For bispanning graphs with vertex-connectivity two, we show that the bispanning graph is the 2-clique sum of two smaller bispanning graphs, and that the unique exchange graph can be built by joining their exchange graphs and forwarding edges at the join seam. And for all remaining bispanning graphs, we prove a composition method at a vertex of degree three, wherein the unique exchange graph is constructed from the exchange graphs of three reduced bispanning graphs.