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International Workshop on Combinatorial Algorithms

IWOCA 2021: Combinatorial Algorithms pp 237–251Cite as

Vertex Cover at Distance on H-Free Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12757))

Abstract

The question of characterizing graphs H such that the Vertex Cover problem is solvable in polynomial time in the class of H-free graphs is notoriously difficult and still widely open. We completely solve the corresponding question for a distance-based generalization of vertex cover called distance-k vertex cover, for any positive integer k. In this problem the task is to determine, given a graph G and an integer \(\ell \), whether G contains a set of at most \(\ell \) vertices such that each edge of G is at distance at most k from a vertex in the set. We show that for all \(k \ge 1\) and all graphs H, the distance-k vertex cover problem is solvable in polynomial time in the class of H-free graphs if H is an induced subgraph of \(P_{2k+2} + sP_{\max \{k,2\}}\) for some \(s \ge 0\), and NP-complete otherwise.

This work is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects J1-9110, N1-0102, N1-0160).

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Acknowledgments

The authors wish to thank Peter Muršič for valuable discussions.

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Authors and Affiliations

  1. FAMNIT, University of Primorska, Koper, Slovenia

    Clément Dallard, Mirza Krbezlija & Martin Milanič

  2. IAM, University of Primorska, Koper, Slovenia

    Clément Dallard & Martin Milanič

Authors
  1. Clément Dallard
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  2. Mirza Krbezlija
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  3. Martin Milanič
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Correspondence to Clément Dallard .

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  1. University of Ottawa, Ottawa, ON, Canada

    Paola Flocchini

  2. University of Ottawa, Ottawa, ON, Canada

    Lucia Moura

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Dallard, C., Krbezlija, M., Milanič, M. (2021). Vertex Cover at Distance on H-Free Graphs. In: Flocchini, P., Moura, L. (eds) Combinatorial Algorithms. IWOCA 2021. Lecture Notes in Computer Science(), vol 12757. Springer, Cham. https://doi.org/10.1007/978-3-030-79987-8_17

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  • DOI: https://doi.org/10.1007/978-3-030-79987-8_17

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