Graphs of Bounded Treewidth can be Canonized in AC1

In recent results the complexity of isomorphism testing on graphs of bounded treewidth is improved to TC1 [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 TC1 was stated as an open question in [K¨ob06]. K¨obler and Verbitsky [KV08] 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 [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.

• Type: Journal
• Year: 2011
• Where: ECCC
• Authors: Fabian Wagner
• Keywords: Canonical, Canonical Labeling, Canonical Labeling Algorithm, ECCC 2011, ECommerce