BOUNDING AND COMPUTING OBSTACLE NUMBERS OF GRAPHS
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10490824" target="_blank" >RIV/00216208:11320/24:10490824 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=iYnpDEu_zj" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=iYnpDEu_zj</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/23M1585088" target="_blank" >10.1137/23M1585088</a>
Alternative languages
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 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 ) [M. Balko, J. Cibulka, and P. Valtr, Discrete Comput. Geom ., 59 (2018), pp. 143--164] and that there are n - vertex graphs whose obstacle number is Cl( n/ (log log n ) 2 ) [V. Dujmovic' and P. Morin, Electron. J. Combin ., 22 (2015), 3.1]. We improve this lower bound to Cl( n/ log log n ) for simple polygons and to Cl( 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 Cl( 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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Classification
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
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
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)
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Discrete Mathematics
ISSN
0895-4801
e-ISSN
1095-7146
Volume of the periodical
38
Issue of the periodical within the volume
2
Country of publishing house
US - UNITED STATES
Number of pages
29
Pages from-to
1537-1565
UT code for WoS article
001232147900004
EID of the result in the Scopus database
2-s2.0-85195195630