# Isomorphism for graphs of bounded distance width

## Abstract

In this paper, we study the Graph Isomorphism problem on graphs of bounded treewidth, bounded degree or bounded bandwidth. There are *O(n*^{c}) algorithms which solve Graph Isomorphism on graphs which have a bound on the treewidth, degree or bandwidth, but the exponent *c* depends on this bound, where *n* is the number of vertices in input graphs. We introduce some new graph parameters: the (rooted) path distance width, which is a restriction of bandwidth, and the (rooted) tree distance width, which is a restriction of treewidth. We give algorithms that solve Graph Isomorphism in *O(n*^{2}) time for graphs with bounded rooted path distance width, and in *O(n*^{3}) time for graphs with bounded rooted tree distance width. Additionally, we show that computing the path distance width of a graph is a NP-complete problem even if the input graphs are restricted to the class of trees. Moreover we show that the rooted path or tree distance width can be computed in *O(ne*) time and both path and tree distance width can be computed in *O(n*^{k+1}) time, when they are bounded by a constant *k*, where e is the number of edges in input graphs. Finally, we study the relationships between the newly introduced parameters and other existing graph parameters.

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