The isomorphism problem for k-trees is complete for logspace.
With V. Arvind, Bireswar Das, Johannes Köbler.
Information and Computation 217:1–11 (Aug. 2012)

Abstract. We show that, for k constant, k-tree isomorphism can be decided in logarithmic space by giving an O(nlogn) 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 O((k+1)!logn), where n is the number of vertices, yielding the fastest known FPT algorithm for k-tree isomorphism.

Conference version:
The isomorphism problem for k-Trees is complete for logspace
With Johannes Köbler.
Mathematical Foundations of Computer Science (Proceedings of 34th MFCS). Springer, 2009. Pp. 537–548.