[1] **viXra:1909.0066 [pdf]**
*submitted on 2019-09-03 12:51:30*

**Authors:** Ortho Flint, Stuart Rankin

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

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.

**Category:** Combinatorics and Graph Theory