TY - GEN

T1 - Bounding and Computing Obstacle Numbers of Graphs

AU - Balko, Martin

AU - Chaplick, Steven

AU - Ganian, Robert

AU - Gupta, Siddharth

AU - Hoffmann, Michael

AU - Valtr, Pavel

AU - Wolff, Alexander

N1 - Publisher Copyright:
© 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

PY - 2022/9/1

Y1 - 2022/9/1

N2 - An obstacle representation of a graph G consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each n-vertex graph is O(n log n) [Balko, Cibulka, and Valtr, 2018] and that there are n-vertex graphs whose obstacle number is Ω(n/(log log n)2) [Dujmovic and Morin, 2015]. We improve this lower bound to Ω(n/ log log n) for simple polygons and to Ω(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmovic and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings Ω(n2) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle.

AB - An obstacle representation of a graph G consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each n-vertex graph is O(n log n) [Balko, Cibulka, and Valtr, 2018] and that there are n-vertex graphs whose obstacle number is Ω(n/(log log n)2) [Dujmovic and Morin, 2015]. We improve this lower bound to Ω(n/ log log n) for simple polygons and to Ω(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmovic and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings Ω(n2) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle.

KW - FPT

KW - NP-hardness

KW - Obstacle number

KW - Obstacle representation

KW - Visibility

UR - http://www.scopus.com/inward/record.url?scp=85137601884&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2022.11

DO - 10.4230/LIPIcs.ESA.2022.11

M3 - Conference contribution

AN - SCOPUS:85137601884

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 30th Annual European Symposium on Algorithms, ESA 2022

A2 - Chechik, Shiri

A2 - Navarro, Gonzalo

A2 - Rotenberg, Eva

A2 - Herman, Grzegorz

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 30th Annual European Symposium on Algorithms, ESA 2022

Y2 - 5 September 2022 through 9 September 2022

ER -