Before revision:  In order to obtain these results, we use the vertex separator of these graphs effectively, and design space-efficient algorithms to find such separators. The constructions of the separators presented here can be of independent interest\log n) space, where n is the number of Jordan regions, and m is the total number of crossings among the regions.
After revision:  In order to obtain these results, we use the vertex separator of these graphs effectively, and design space-efficient manner and use it to solve the Reachability in polynomial time and O(m^{1/2\log n) space, where n is the number of Jordan regions, and m is the total number of crossings among the regions.