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Maximum Cut on Interval Graphs of Interval Count Four is NP-Complete

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Abstract

The computational complexity of the MaxCut problem restricted to interval graphs has been open since the 80’s, being one of the problems proposed by Johnson in his Ongoing Guide to NP-completeness, and has been settled as NP-complete only recently by Adhikary, Bose, Mukherjee, and Roy. On the other hand, many flawed proofs of polynomiality for MaxCut on the more restrictive class of unit/proper interval graphs (or graphs with interval count 1) have been presented along the years, and the classification of the problem is still unknown. In this paper, we present the first NP-completeness proof for MaxCut when restricted to interval graphs with bounded interval count, namely graphs with interval count 4.

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Acknowledgements

We thank Vinicius F. Santos who shared reference [1], and anonymous referees for many valuable suggestions, including improving the interval count from 5 to 4.

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Authors and Affiliations

  1. Programa de Engenharia de Sistemas e Computção, Universidade Federal do Rio de Janeiro, Caixa Postal 68511, Rio de Janeiro, Rio de Janeiro, 21941-972, Brazil

    Celina M. H. de Figueiredo & Alexsander A. de Melo

  2. Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier, 524, Rio de Janeiro, Rio de Janeiro, 20550-900, Brazil

    Fabiano S. Oliveira

  3. Departamento de Matemática - Centro de Ciências, Universidade Federal do Ceará, Av. Mister Hull s/n - Bloco 914, Fortaleza, Ceará, 60455-760, Brazil

    Ana Silva

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  1. Celina M. H. de Figueiredo
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  2. Alexsander A. de Melo
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Correspondence to Ana Silva.

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de Figueiredo, C.M.H., de Melo, A.A., Oliveira, F.S. et al. Maximum Cut on Interval Graphs of Interval Count Four is NP-Complete. Discrete Comput Geom 71, 893–917 (2024). https://doi.org/10.1007/s00454-023-00508-x

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