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Geodesic Obstacle Representation of Graphs

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Abstract

An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the resulting representation is a plane straight-line embedding of the graph) have been extensively studied with the main objective of minimizing the number of obstacles. Recently, Biedl and Mehrabi [Therese C. Biedl and Saeed Mehrabi, 2017] studied non-blocking grid obstacle representations of graphs in which the vertices of the graph are mapped onto points in the plane while the straight-line segments representing the adjacency between the vertices is replaced by the L_1 (Manhattan) shortest paths in the plane that avoid obstacles.
In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in R^d as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such representations.

BibTeX - Entry

```@InProceedings{bose_et_al:LIPIcs:2018:9027,
author =	{Prosenjit Bose and Paz Carmi and Vida Dujmovic and Saeed Mehrabi and Fabrizio Montecchiani and Pat Morin and Luis Fernando Schultz Xavier da Silveira},
title =	{{Geodesic Obstacle Representation of Graphs}},
booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
pages =	{23:1--23:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-076-7},
ISSN =	{1868-8969},
year =	{2018},
volume =	{107},
editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},