  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 are inspired by and generalise 'partition
backtrack', which is the current state-of-the-art algorithm introduced by
Jeffrey Leon in 1991. But, instead of ordered partitions, we use labelled
directed graphs to organise our backtrack search algorithms, which allows for a
richer representation of many problems while often resulting in smaller search
spaces. In this article we present the theory underpinning our framework, we
describe our algorithms, and we show the results of some experiments. An
implementation of our algorithms is available as free software in the
GraphBacktracking package for GAP.
