  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.
