The Isomorphism Problem for k-Trees Is Complete for Logspace

We show that, for k constant, k-tree isomorphism can be decided in logarithmic space by giving a logspace canonical labeling algorithm. The algorithm computes a unique tree decomposition, uses colors to fully encode the structure of the original graph in the decomposition tree and invokes Lindell’s tree canonization algorithm. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k-trees are all complete for deterministic logspace. Completeness for logspace holds even for simple structural properties of k-trees. We also show that isomorphism of all k-trees, k ≥ 1, is fixed parameter tractable with respect to k by giving an algorithm running in time O((k + 2)! ·m), where m is the number of edges in the input graph.