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Large independent sets in triangle-free planar graphs
Publikationstyp
Conference Paper
Date Issued
2017
Sprache
English
Author(s)
TORE-URI
Volume
31
Issue
2
Start Page
1355
End Page
1373
Citation
SIAM Journal on Discrete Mathematics (2017)
Publisher DOI
ArXiv ID
Publisher
Society for Industrial and Applied Mathematics
Every triangle-free planar graph on n vertices has an independent set of size at least (n + 1)/3, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph G on n vertices and an integer k ≥ 0, decides whether G has an independent set of size at least (n + k)/3, in time 2O(√k)n. Thus, the problem is fixed-parameter tractable when parameterized by k. Furthermore, as a corollary of the result used to prove the correctness of the algorithm, we show that there exists ϵ > 0 such that every planar graph of girth at least five on n vertices has an independent set of size at least n/(3-ϵ). We further give an algorithm that, given a planar graph G of maximum degree 4 on n vertices and an integer k ≥ 0, decides whether G has an independent set of size at least (n + k)/4, in time 2O(√k)n. 7copy; 2017 Zdeňek Dvořák, Matthias Mnich.
Subjects
Fixed-parameter tractability
Independent set
Planar graphs
Treewidth