Canonical decompositions and algorithmic recognition of spatial graphs

TL;DR

This paper presents an algorithm to determine isomorphism of spatial graphs, including those with additional vertex and edge colorings or orientations, by canonical decomposition into non-separable blocks.

Contribution

It introduces a method for canonical decomposition of spatial graphs and applies existing algorithms to recognize isomorphism in a generalized setting.

Findings

Algorithm exists for spatial graph isomorphism with colorings and orientations

Spatial graphs can be decomposed into canonical blocks for analysis

The approach extends recognition algorithms to more complex spatial graphs

Abstract

We prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colorings, edge colorings and/or edge orientations. We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-separable and have no cut vertices, in a suitable topological sense. Then we apply a result of Haken and Matveev in order to algorithmically distinguish these blocks.