Large Independent Sets in Triangle-Free Planar Graphs

Zdenek Dvorak, Matthias Mnich

    Research output: Contribution to journalArticleAcademicpeer-review

    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 languageEnglish
    Pages (from-to)1355-1373
    Number of pages19
    JournalSiam Journal on Discrete Mathematics
    Volume31
    Issue number2
    DOIs
    Publication statusPublished - Jun 2017

    Keywords

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

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