Computing Bounded Path Decompositions in Logspace

The complexity of Graph Isomorphism (GI) is a long-standing open problem. GI is not known to be solvable in polynomial time and is unlikely to be NP-complete [BHZ87, Sch88]. Polynomial-time algorithms are known for GI of several special classes of graphs. One such interesting class is graphs excluding a fixed minor. Any family of graphs that is closed under taking minors falls into this class. Important examples of such families include bounded genus graphs, bounded pathwidth graphs and bounded treewidth graphs. Recent papers proved the logspace-completeness of GI of planar graphs and graphs of treewidth at most three, thus settling their complexity [ADK08, DLN09, DNTW09]. These algorithms are based on computing certain kind of decompositions in logspace. The best known upper bound for GI of bounded treewidth graphs is LogCFL [DTW10], implying the same upper bound for graphs of bounded pathwidth. One bottleneck towards improving these upper bounds is computing bounded width tree/path decompositions in logspace. Elberfeld, Jakoby and Tantau [EJT10] presented a logspace algorithm to compute tree decompositions of bounded treewidth graphs. In this paper, we present a logspace algorithm to compute path decompositions of bounded pathwidth graphs, thus removing a bottleneck towards settling the complexity of GI of bounded pathwidth graphs. Prior to our work, the best known upper bound to compute such decompositions is polynomial time [BK96].