 
Summary: Planar and Grid Graph Reachability Problems
Eric Allender
David A. Mix Barrington
Tanmoy Chakraborty
Samir Datta§
Sambuddha Roy¶
Abstract
We study the complexity of restricted versions of stconnectivity, which is the standard complete problem for 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 logspaceequivalent to reachability in grid graphs (and in particular it is
logspaceequivalent to both reachability and nonreachability in planar graphs).
· Many of the natural restrictions on gridgraph reachability (GGR) are equivalent under AC0
reductions (for instance,
undirected GGR, outdegreeone GGR, and indegreeoneoutdegreeone 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 [BK78]. These problems provide natural
examples of problems that are hard for NC1
under AC0
reductions but are not known to be hard for L; they thus give
