TY - JOUR

T1 - Geodesic obstacle representation of graphs

AU - Bose, Prosenjit

AU - Carmi, Paz

AU - Dujmović, Vida

AU - Mehrabi, Saeed

AU - Montecchiani, Fabrizio

AU - Morin, Pat

AU - Xavier da Silveira, Luís Fernando Schultz

N1 - Funding Information:
A preliminary version of this paper appeared in proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) [7] . Research of Prosenjit Bose, Vida Dujmović, and Pat Morin is supported by Natural Sciences and Engineering Research Council of Canada (NSERC). Saeed Mehrabi is supported by a Carleton-Fields postdoctoral fellowship . Research of Fabrizio Montecchiani partially supported by Dipartimento di Ingegneria, Università degli Studi di Perugia , RICBA19FM : “Modelli, algoritmi e sistemi per la visualizzazione di grafi e reti”.
Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2023/2/1

Y1 - 2023/2/1

N2 - 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. Recently, Biedl and Mehrabi (Graph Drawing 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 are replaced by the L1 (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 Rd 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.

AB - 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. Recently, Biedl and Mehrabi (Graph Drawing 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 are replaced by the L1 (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 Rd 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.

KW - Geodesic obstacle representation

KW - Grid obstacle representation

KW - Obstacle representation

UR - http://www.scopus.com/inward/record.url?scp=85139348473&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2022.101946

DO - 10.1016/j.comgeo.2022.101946

M3 - Article

AN - SCOPUS:85139348473

VL - 109

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

M1 - 101946

ER -