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Computing L(p, 1)-Labeling with Combined Parameters

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WALCOM: Algorithms and Computation (WALCOM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12635))

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Abstract

Given a graph, an L(p, 1)-labeling of the graph is an assignment f from the vertex set to the set of nonnegative integers such that for any pair of vertices \((u,v),|f (u) - f (v)| \ge p\) if u and v are adjacent, and \(f(u) \ne f(v)\) if u and v are at distance 2. The L(p, 1)-labeling problem is to minimize the span of f (i.e.,\(\max _{u\in V}(f(u)) - \min _{u\in V}(f(u))+1\)). It is known to be NP-hard even for graphs of maximum degree 3 or graphs with tree-width 2, whereas it is fixed-parameter tractable with respect to vertex cover number. Since the vertex cover number is a kind of the strongest parameter, there is a large gap between tractability and intractability from the viewpoint of parameterization. To fill up the gap, in this paper, we propose new fixed-parameter algorithms for L(p, 1)-Labeling by the twin cover number plus the maximum clique size and by the tree-width plus the maximum degree. These algorithms reduce the gap in terms of several combinations of parameters.

This work is partially supported by JSPS KAKENHI Grant Numbers JP17K19960, JP17H01698, JP19K21537 and JP20H05967. A full version is available in [21].

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Acknowledgements

We are grateful to Dr. Yota Otachi for his insightful comments.

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Authors and Affiliations

  1. Chuo University, Tokyo, 112-8551, Japan

    Tesshu Hanaka

  2. Nagoya University, Nagoya, 464-8601, Japan

    Kazuma Kawai & Hirotaka Ono

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  1. Tesshu Hanaka
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  2. Kazuma Kawai
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  3. Hirotaka Ono
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Correspondence to Tesshu Hanaka .

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Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Ishikawa, Japan

    Ryuhei Uehara

  2. University of Sydney, Sydney, NSW, Australia

    Seok-Hee Hong

  3. Indian Statistical Institute, kolkata, India

    Subhas C. Nandy

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Hanaka, T., Kawai, K., Ono, H. (2021). Computing L(p, 1)-Labeling with Combined Parameters. In: Uehara, R., Hong, SH., Nandy, S.C. (eds) WALCOM: Algorithms and Computation. WALCOM 2021. Lecture Notes in Computer Science(), vol 12635. Springer, Cham. https://doi.org/10.1007/978-3-030-68211-8_17

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  • DOI: https://doi.org/10.1007/978-3-030-68211-8_17

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