The isomorphism problem for k-trees is complete for logspace

Abstract

We show that, for k constant, k -tree isomorphism can be decided in logarithmic space by giving an View the MathML sourceO(klogn) space 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 Lindellʼs 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 a variant of our canonical labeling algorithm runs in time View the MathML sourceO((k+1)!n), where n is the number of vertices, yielding the fastest known FPT algorithm for k-tree isomorphism.