# Space-Efficient Algorithms for Reachability in Directed Geometric Graphs

Sujoy Bhore, Rahul Jain

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## 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(m1/2 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(m1/2 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(n1/4+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.

Original language English 32nd International Symposium on Algorithms and Computation, ISAAC 2021 Hee-Kap Ahn, Kunihiko Sadakane Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing 9783959772143 https://doi.org/10.4230/LIPIcs.ISAAC.2021.63 Published - 1 Dec 2021 Yes 32nd International Symposium on Algorithms and Computation, ISAAC 2021 - Fukuoka, JapanDuration: 6 Dec 2021 → 8 Dec 2021

### Publication series

Name Leibniz International Proceedings in Informatics, LIPIcs 212 1868-8969

### Conference

Conference 32nd International Symposium on Algorithms and Computation, ISAAC 2021 Japan Fukuoka 6/12/21 → 8/12/21

## Keywords

• Geometric intersection graphs
• Reachablity
• Space-efficient algorithms

• Software

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