### Abstract

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 2(O(root 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 epsilon > 0 such that every planar graph of girth at least five on n vertices has an independent set of size at least n/(3-epsilon). 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 2(O(root k)) n.

Original language | English |
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Pages (from-to) | 1355-1373 |

Number of pages | 19 |

Journal | Siam Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2017 |

### Keywords

- planar graphs
- independent set
- fixed-parameter tractability
- treewidth
- 1ST-ORDER PROPERTIES
- BOUNDED EXPANSION
- ALGORITHM
- NUMBER
- GIRTH
- GRAD

## Cite this

*Siam Journal on Discrete Mathematics*,

*31*(2), 1355-1373. https://doi.org/10.1137/16M1061862