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URN: urn:nbn:de:0030-drops-154961
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### Space-Efficient Algorithms for Reachability in Directed Geometric Graphs

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### Abstract

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, 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.

### BibTeX - Entry

```@InProceedings{bhore_et_al:LIPIcs.ISAAC.2021.63,
author =	{Bhore, Sujoy and Jain, Rahul},
title =	{{Space-Efficient Algorithms for Reachability in Directed Geometric Graphs}},
booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
pages =	{63:1--63:17},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-214-3},
ISSN =	{1868-8969},
year =	{2021},
volume =	{212},
editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/15496},
URN =		{urn:nbn:de:0030-drops-154961},
doi =		{10.4230/LIPIcs.ISAAC.2021.63},
annote =	{Keywords: Reachablity, Geometric intersection graphs, Space-efficient algorithms}
}```

 Keywords: Reachablity, Geometric intersection graphs, Space-efficient algorithms Seminar: 32nd International Symposium on Algorithms and Computation (ISAAC 2021) Issue date: 2021 Date of publication: 30.11.2021

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