Reachability in K3,3-free and K5-free Graphs is in Unambiguous Logspace

We show that the reachability problem for directed graphs that are either K3,3-free or K5-free is in unambiguous log-space, UL∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL∩ coUL. Our algorithm decomposes the graphs into biconnected and triconnected components. This gives a tree structure on these components. The non-planar components are replaced by planar components that maintain the reachability properties. For K5-free graphs we also need a decomposition into 4-connected components. Thereby we provide a logspace reduction to the planar reachability problem. We show the same upper bound for computing distances in K3,3-free and K5-free directed graphs and for computing longest paths in K3,3-free and K5-free directed acyclic graphs.