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Complexity of Fall Coloring for Restricted Graph Classes

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Abstract

We strengthen a result by Laskar and Lyle (Discrete Appl. Math. 157, 330–338 2009) by proving that it is NP-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding whether a bipartite graph can be partitioned into k independent dominating sets is NP-complete for every k ≥ 3. We also strengthen a result by Henning et al. (Discrete Math. 309(23), 6451–6458 2009) by showing that it is NP-complete to determine if a graph has two disjoint independent dominating sets, even when the problem is restricted to triangle-free planar graphs. Finally, for every k ≥ 3, we show that there is some constant t depending only on k such that deciding whether a k-regular graph can be partitioned into t independent dominating sets is NP-complete. We conclude by deriving moderately exponential-time algorithms for the problem.

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Authors and Affiliations

  1. Helsinki, Finland

    Juho Lauri

  2. University of Cyprus, Nicosia, Cyprus

    Christodoulos Mitillos

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  1. Juho Lauri
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  2. Christodoulos Mitillos
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Correspondence to Juho Lauri.

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This article belongs to the Topical Collection: Special Issue on International Workshop on Combinatorial Algorithms (IWOCA 2019)

Guest Editors: Charles Colbourn, Roberto Grossi, Nadia Pisanti

Part of this work was done while J.L. was at Nokia Bell Labs, Ireland

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Lauri, J., Mitillos, C. Complexity of Fall Coloring for Restricted Graph Classes. Theory Comput Syst 64, 1183–1196 (2020). https://doi.org/10.1007/s00224-020-09982-9

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