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The Complexity of L(p, q)-Edge-Labelling

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Abstract

The L(p, q)-Edge-Labelling problem is the edge variant of the well-known L(p, q)-Labelling problem. It is equivalent to the L(p, q)-Labelling problem itself if we restrict the input of the latter problem to line graphs. So far, the complexity of L(p, q)-Edge-Labelling was only partially classified in the literature. We complete this study for all \(p,q\ge 0\) by showing that whenever \((p,q)\ne (0,0)\), the L(p, q)-Edge-Labelling problem is NP-complete. We do this by proving that for all \(p,q\ge 0\) except \(p=q=0\), there is an integer k so that L (p ,  q)-Edge-k -Labelling is NP-complete.

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Notes

  1. See http://wwwusers.di.uniroma1.it/~calamo/survey.html for later results.

  2. The Python program for checking this can be found in the following github repository: https://github.com/G-Berthe/Lpq-edge-labelling/ Use, e.g.: extended_four_star=[(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,7),(4,8)] with plotPoss(extended_four_star, 3, 2, 12, {(1,5):[2]}) to find \(2: \{2,3,9\}\).

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Acknowledgements

An extended abstract of this paper, omitting numerous proofs, appeared at The 16th International Conference and Workshops on Algorithms and Computation (WALCOM) 2022 [1].

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Authors and Affiliations

  1. ENS de Lyon, Lyon, France

    Gaétan Berthe

  2. Department of Computer Science, Durham University, Durham, UK

    Barnaby Martin & Daniël Paulusma

  3. University of Bristol, Bristol, UK

    Siani Smith

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  1. Gaétan Berthe
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  2. Barnaby Martin
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  3. Daniël Paulusma
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  4. Siani Smith
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Correspondence to Barnaby Martin.

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Berthe, G., Martin, B., Paulusma, D. et al. The Complexity of L(p, q)-Edge-Labelling. Algorithmica 85, 3406–3429 (2023). https://doi.org/10.1007/s00453-023-01120-4

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