# Project description

We define a binary coupling tree on *n+1* leaves as an
unordered binary tree in which each leaf has a distinct label. A
Yutsis graph of order *n* consists of two binary coupling
trees on *n+1* leaves, using the same label set, in which the
unique leaves with the same label are identified. Note that both
root nodes are also connected by an edge. The leaf nodes
themselves disappear from the graph.

The following figure shows an example of two binary coupling
trees (a), the corresponding Yutsis graph of order 4 (b) and the
same Yutsis graph redrawn (c).

The motivation for this definition comes from theoretical
physics. In the context of quantum theory of angular momentum
Yutsis graphs are used to represent a *3nj*-coefficient,
while the binary coupling trees correspond to the coupling schemes
in the bra/kets of the *3nj*-coefficient (see [BL81] and [YLV62]).

A Yutsis graph of order *n* is a cubic graph with *2n
*nodes and *3n* edges, with the additional property that
it contains an edge-cut on *n+2* edges separating the graph
into two vertex-induced trees of equal size. We call such a cut a
defining cut of the Yutsis graph, the two trees are called
defining trees of the Yutsis graph. By construction Yutsis graphs
cannot contain a bridge.

In this project we tackle two
problems:

- Classification of Yutsis graphs:
which cubic graphs are Yutsis graphs?
- Reduction of Yutsis graphs:
Find an optimal, or at least an efficient,
reduction for a given Yutsis graph.