Space-efficient algorithms for reachability in directed geometric graphs

The problem of graph REACHABILITY is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the REACHABILITY problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the REACHABILITY problem. For intersection graphs of Jordan regions, we show how to obtain a “good” vertex separator in a space-efficient manner and use it to solve the REACHABILITY in polynomial time and O(m^1/2log⁡n) space, where n is the number of Jordan regions, and m is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial time and O(m^1/2log⁡n) space algorithm, where n and m are the number of vertices and edges, respectively. However, for unit contact disk graphs (penny graphs), we use a more involved technique and obtain a better algorithm. We show that for every ϵ>0, there exists a polynomial time algorithm that can solve REACHABILITY in an n vertex directed penny graph, using O(n^1/4+ϵ) space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques.