Planar and Grid Graph Reachability Problems

AbstractWe study the complexity of restricted versions of s-t-connectivity, which is the standard complete problem for $\mathsf{NL}$ . In particular, we focus on different classes of planar graphs, of which grid graphs are an important special case. Our main results are: Reachability in graphs of genus one is logspace-equivalent to reachability in grid graphs (and in particular it is logspace-equivalent to both reachability and non-reachability in planar graphs).Many of the natural restrictions on grid-graph reachability (GGR) are equivalent under $\mathsf{AC}^{0}$ reductions (for instance, undirected GGR, outdegree-one GGR, and indegree-one-outdegree-one GGR are all equivalent). These problems are all equivalent to the problem of determining whether a completed game position in HEX is a winning position, as well as to the problem of reachability in mazes studied by Blum and Kozen (IEEE Symposium on Foundations of Computer Science (FOCS), pp. 132–142, [1978]). These problems provide natural examples of problems that are hard for $\mathsf{NC}^{1}$ under $\mathsf{AC}^{0}$ reductions but are not known to be hard for  $\mathsf{L}$ ; they thus give insight into the structure of $\mathsf{L}$ .Reachability in layered planar graphs is logspace-equivalent to layered grid graph reachability (LGGR). We show that LGGR lies in $\mathsf{UL}$ (a subclass of $\mathsf{NL}$ ).Series-Parallel digraphs (on which reachability was shown to be decidable in logspace by Jakoby et al.) are a special case of single-source-single-sink planar directed acyclic graphs (DAGs); reachability for such graphs logspace reduces to single-source-single-sink acyclic grid graphs. We show that reachability on such grid graphs $\mathsf{AC}^{0}$ reduces to undirected GGR.We build on this to show that reachability for single-source multiple-sink planar DAGs is solvable in $\mathsf{L}$ .