Graphs of Bounded Treewidth can be Canonized in AC

In recent results the complexity of isomorphism testing on graphs of bounded treewidth is improved to TC [GV06] and further to LogCFL [DTW10]. The computation of canonical forms or a canonical labeling provides more information than isomorphism testing. Whether canonization is in NC or even TC was stated as an open question in [Köb06]. Köbler and Verbitsky [KV08] give a TC canonical labeling algorithm. We show that a canonical labeling can be computed in AC. This is based on several ideas, e.g. that approximate tree decompositions of logarithmic depth can be obtained in logspace [EJT10a], and techniques of Lindells tree canonization algorithm [Lin92]. We define recursively what we call a minimal description which gives with respect to some parameters in a logarithmic number of levels a canonical invariant together with an arrangement of all vertices. From this we compute a canonical labeling.