McKay’s Canonical Graph Labeling Algorithm

The problem of deciding whether two graphs are isomorphic is fundamental in graph theory. Moreover, the flexibility with which other combinatorial objects can be modeled by graphs has meant that efficient programs for deciding whether graphs are isomorphic have also been used to study a variety of other combinatorial structures. Not only is the graph isomorphism problem a very practical one, it is also fascinating from a complexity-theoretic point of view. Graph isomorphism is one of the few problems that are clearly in NP but not known either to be solvable in polynomial time, or to be NPcomplete. Various people have worked to create algorithms for graph isomorphism which are “practical in practice”. One of the most powerful and best known of these algorithms is due to Brendan McKay. It is known that his algorithm has exponential running time on some inputs, but it performs exceptionally well under most circumstances. In this article we aim to provide an introduction to the essential ideas of McKay’s algorithm.