Drawing Graphs Using a Small Number of Obstacles

Result code in 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>
Result on the web
<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>

Result language
angličtina
Original language name
Drawing Graphs Using a Small Number of Obstacles
Original language description
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<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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Center of excellence - Institute for theoretical computer science (CE-ITI)</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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