A Characterization of Individualization-Refinement Trees

TL;DR

This paper provides a precise characterization of the structure of individualization-refinement trees used in graph symmetry algorithms, including conditions for when a tree can be realized by a graph.

Contribution

It offers a constructive, necessary and sufficient characterization of IR-trees, applicable to both colored and uncolored trees, with a method to generate graphs from trees.

Findings

Characterization of IR-trees as necessary and sufficient

Construction method for graphs from IR-trees

Applicable to both colored and uncolored trees

Abstract

Individualization-Refinement (IR) algorithms form the standard method and currently the only practical method for symmetry computations of graphs and combinatorial objects in general. Through backtracking, on each graph an IR-algorithm implicitly creates an IR-tree whose order is the determining factor of the running time of the algorithm. We give a precise and constructive characterization which trees are IR-trees. This characterization is applicable both when the tree is regarded as an uncolored object but also when regarded as a colored object where vertex colors stem from a node invariant. We also provide a construction that given a tree produces a corresponding graph whenever possible. This provides a constructive proof that our necessary conditions are also sufficient for the characterization.