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Cutting Barnette Graphs Perfectly is Hard

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Graph-Theoretic Concepts in Computer Science (WG 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14093))

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Abstract

A perfect matching cut is a perfect matching that is also a cutset, or equivalently a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be NP-complete in subcubic bipartite graphs [Le & Telle, TCS ’22] but its complexity was open in planar graphs and in cubic graphs. We settle both questions at once by showing that Perfect Matching Cut is NP-complete in 3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our work, among problems whose input is solely an undirected graph, only Distance-2 4-Coloring was known NP-complete in Barnette graphs. Notably, Hamiltonian Cycle would only join this private club if Barnette’s conjecture were refuted.

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Notes

  1. 1.

    Note however that the induced variant of Maximum Matching is an interesting problem that happens to be NP-complete [36].

  2. 2.

    The authors consider the framework of \((k,\sigma ,\rho )\)-partition problem, where k is a positive integer, and \(\sigma , \rho \) are sets of non-negative integers, and one looks for a vertex-partition into k parts such that each vertex of each part has a number of neighbors in its own part in \(\sigma \), and a number of other neighbors in \(\rho \); hence, PMC is then the \((2,\mathbb {N},\{1\})\)-partition problem.

  3. 3.

    Among problems with edge orientations, vertex or edge weights, or prescribed subsets of vertices or edges, the list is significantly longer, and also includes Minimum Weighted Edge Coloring [7], List Edge Coloring and Precoloring Extension [30], k-In-A-Tree [8], etc.

  4. 4.

    Which precisely states that every polyhedral (that is, 3-connected planar) cubic bipartite graphs admits a hamiltonian cycle.

  5. 5.

    We avoid using the term “edge cut” since, for some authors, an edge cut is, more generally, a subset of edges whose deletion increases the number of connected components.

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Acknowledgments

We are much indebted to Carl Feghali for introducing us to the topic of (perfect) matching cuts, and presenting us with open problems that led to the current paper. We also wish to thank him and Kristóf Huszár for helpful discussions on an early stage of the project.

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Authors and Affiliations

  1. Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, Lyon, France

    Édouard Bonnet, Dibyayan Chakraborty & Julien Duron

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  1. Édouard Bonnet
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  2. Dibyayan Chakraborty
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  3. Julien Duron
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Correspondence to Dibyayan Chakraborty .

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Editors and Affiliations

  1. Durham University, Durham, UK

    Daniël Paulusma

  2. University of Fribourg, Fribourg, Fribourg, Switzerland

    Bernard Ries

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Bonnet, É., Chakraborty, D., Duron, J. (2023). Cutting Barnette Graphs Perfectly is Hard. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_9

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