Graphs of Bounded Treewidth Can Be Canonized in AC1 (2011)
Link: https://link.springer.com/chapter/10.1007/978-3-642-20712-9_16
Abstract: In recent results the complexity of isomorphism testing on graphs of bounded treewidth is improved to TC1 [17] and further to LogCFL [11]. The computation of canonical forms or a canonical labeling provides more information than isomorphism testing. Whether canonization is in NC or even TC1 was stated as an open question in [18]. Köbler and Verbitsky [20] give a TC2 canonical labeling algorithm. We show that a canonical labeling can be computed in AC1. This is based on several ideas, e.g. that approximate tree decompositions of logarithmic depth can be obtained in logspace [15], and techniques of Lindells tree canonization algorithm [22]. 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.