  We introduce a new framework for solving an important class of computational
problems involving finite permutation groups, which includes calculating set
stabilisers, intersections of subgroups, and isomorphisms of combinatorial
structures. Our techniques generalise 'partition backtrack', which is the
current state-of-the-art algorithm introduced by Jeffrey Leon in 1991, and
which has inspired our work. Our backtrack search algorithms are organised
around vertex- and arc-labelled directed graphs, which allow us to represent
many problems more richly than do ordered partitions. We present the theory
underpinning our framework, and we include the results of experiments showing
that our techniques often result in smaller search spaces than does partition
backtrack. An implementation of our algorithms is available as free software in
the GraphBacktracking package for GAP.
