## Combinatorics and Graph Theory   ## The Labelled Cycle-Decomposition Trees of a Connected Graph

Authors: Ortho Flint, Stuart Rankin

The notion of a labelled cycle-decomposition tree for an arbitrary graph is introduced in this paper. The idea behind the labelled cycle-decomposition tree, one constructed for each vertex in the graph, is to attach to each vertex a data structure that gives more than local information about the vertex, where the data structures can be compared in polynomial time. The fact that trees can be compared in linear time, with particularly efficient algorithms for solving the isomorphism problem for rooted and labelled trees, led us to consider how we could represent the vertex's view of the graph by means of a labelled rooted tree. Of course, cycles in the graph can't be directly recorded by means of a tree structure, but in this article, we present one method for recording certain of the cycles that are encountered during a breadth-first search from the vertex in question. The collection of labelled, so-called cycle-decomposition trees for the graph, one for each vertex, provide an invariant of the graph.

Comments: 10 Pages. The labelled cycle-decomposition trees are a powerful invariant and computationally inexpensive to produce.