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. 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. We use a similar approach for chordal graphs and obtain a polynomial-time and O(m^{1/2}\log n) space algorithm, where n and m are the number of vertices and edges, respectively. However, we use a more involved technique for unit contact disk graphs (penny graphs) and obtain a better algorithm. We show that for every \epsilon> 0, there exists a polynomial-time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n^{1/4+\epsilon}) 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} .
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^{1/2\log 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^{1/2}\log n) space algorithm, where n and m are the number of vertices and edges, respectively. However, we use a more involved technique for unit contact disk graphs (penny graphs) and obtain a better algorithm. We show that for every \epsilon> 0, there exists a polynomial-time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n^{1/4+\epsilon}) 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} .
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In order to obtain these results, we use the vertex separator of these graphs effectively, and design
For intersection graphs of Jordan regions, we show how to obtain a "good" vertex separator in a
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algorithms to find such separators. The constructions of the separators presented here can be of independent interest
manner and use it to solve the Reachability in polynomial time and O(m^{1/2