Bounding and Computing Obstacle Numbers of Graphs

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Bounding and Computing Obstacle Numbers of Graphs

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Bounding and Computing Obstacle Numbers of Graphs

Popis výsledku anglicky

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.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

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

Projekt

<a href="/cs/project/GA21-32817S" target="_blank" >GA21-32817S: Algoritmické, strukturální a složitostní aspekty geometrických konfigurací</a><br>

Návaznosti

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

Rok uplatnění

2022

Kód důvěrnosti údajů

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

Název statě ve sborníku

Leibniz International Proceedings in Informatics, LIPIcs

ISBN

978-3-95977-247-1

ISSN

1868-8969

e-ISSN

—

Počet stran výsledku

13

Strana od-do

—

Název nakladatele

Schloss Dagstuhl

Místo vydání

Německo

Místo konání akce

Postupim

Datum konání akce

5. 9. 2022

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

—