Drawing Graphs Using a Small Number of Obstacles

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10366667" target="_blank" >RIV/00216208:11320/18:10366667 - isvavai.cz</a>

Výsledek na webu

<a href="https://link.springer.com/article/10.1007%2Fs00454-017-9919-2" target="_blank" >https://link.springer.com/article/10.1007%2Fs00454-017-9919-2</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00454-017-9919-2" target="_blank" >10.1007/s00454-017-9919-2</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Drawing Graphs Using a Small Number of Obstacles

Popis výsledku v původním jazyce

An obstacle representation of a graph G is a set of points in the plane representing the vertices of G, together with a set of polygonal obstacles such that two vertices of G are connected by an edge in G if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number obs(G) of G is the minimum number of obstacles in an obstacle representation of G. We provide the first non-trivial general upper bound on the obstacle number of graphs by showing that every n-vertex graph G satisfies obs(G)LESS-THAN OR EQUAL TOnLEFT CEILINGlognRIGHT CEILINGMINUS SIGN n+1. This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. For n-vertex graphs with bounded chromatic number, we improve this bound to O(n). Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound 2Ω(hn) on the number of n-vertex graphs with obstacle number at most h for h&lt;n and a lower bound Ω(n^(4/3)M^(2/3)) for the complexity of a collection of MGREATER-THAN OR EQUAL TOΩ(nlog^(3/2)(n)) faces in an arrangement of line segments with n endpoints. The latter bound is tight up to a multiplicative constant.

Název v anglickém jazyce

Drawing Graphs Using a Small Number of Obstacles

Popis výsledku anglicky

An obstacle representation of a graph G is a set of points in the plane representing the vertices of G, together with a set of polygonal obstacles such that two vertices of G are connected by an edge in G if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number obs(G) of G is the minimum number of obstacles in an obstacle representation of G. We provide the first non-trivial general upper bound on the obstacle number of graphs by showing that every n-vertex graph G satisfies obs(G)LESS-THAN OR EQUAL TOnLEFT CEILINGlognRIGHT CEILINGMINUS SIGN n+1. This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. For n-vertex graphs with bounded chromatic number, we improve this bound to O(n). Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound 2Ω(hn) on the number of n-vertex graphs with obstacle number at most h for h&lt;n and a lower bound Ω(n^(4/3)M^(2/3)) for the complexity of a collection of MGREATER-THAN OR EQUAL TOΩ(nlog^(3/2)(n)) faces in an arrangement of line segments with n endpoints. The latter bound is tight up to a multiplicative constant.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Centrum excelence - Institut teoretické informatiky (CE-ITI)</a><br>

Návaznosti

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

Rok uplatnění

2018

Kód důvěrnosti údajů

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

Název periodika

Discrete and Computational Geometry

ISSN

0179-5376

e-ISSN

—

Svazek periodika

2018

Číslo periodika v rámci svazku

59

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

22

Strana od-do

143-164

Kód UT WoS článku

000418291200006

EID výsledku v databázi Scopus

2-s2.0-85027833996