Bounding and Computing Obstacle Numbers of Graphs

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10453250" target="_blank" >RIV/00216208:11320/22:10453250 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.4230/LIPIcs.ESA.2022.11" target="_blank" >https://doi.org/10.4230/LIPIcs.ESA.2022.11</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.ESA.2022.11" target="_blank" >10.4230/LIPIcs.ESA.2022.11</a>

Result language

angličtina

Original language name

Bounding and Computing Obstacle Numbers of Graphs

Original language description

An obstacle representation of a graph~$G$ consists of a set of pairwise disjoint simply-connected closed regions % in the plane 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 $Omega(n/(loglog n)^2)$ [Dujmovi&apos;c and Morin, 2015]. We improve this lower bound to $Omega(n/loglog n)$ for simple polygons and to $Omega(n)$ 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 Dujmovi&apos;c 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 $Omega(n^2)$ 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.

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

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OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

<a href="/en/project/GA21-32817S" target="_blank" >GA21-32817S: Algorithmic, structural and complexity aspects of geometric configurations</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2022

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

Leibniz International Proceedings in Informatics, LIPIcs

ISBN

978-3-95977-247-1

ISSN

1868-8969

e-ISSN

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Number of pages

13

Pages from-to

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Publisher name

Schloss Dagstuhl

Place of publication

Německo

Event location

Postupim

Event date

Sep 5, 2022

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

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