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A polynomial-time algorithm of finding a minimum k-path vertex cover and a maximum k-path packing in some graphs

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Abstract

For a graph G and a positive integer k, a subset C of vertices of G is called a k-path vertex cover if C intersects all paths of k vertices in G. The cardinality of a minimum k-path vertex cover is denoted by \(\beta _{P_k}(G)\). For a graph G and a positive integer k, a subset M of pairwise vertex-disjoint paths of k vertices in G is called a k-path packing. The cardinality of a maximum k-path packing is denoted by \(\mu _{P_k}(G)\). In this paper, we describe some graphs, having equal values of \(\beta _{P_k}\) and \(\mu _{P_k}\), for \(k \ge 5\), and present polynomial-time algorithms of finding a minimum k-path vertex cover and a maximum k-path packing in such graphs.

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Acknowledgements

The results of Sect. 2 were prepared within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE). The results of Sect. 3 were obtained with the support of the RFFI Grant 18-31-20001-mol-a-ved.

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Authors and Affiliations

  1. National Research Lobachevsky State University of Nizhni Novgorod, 23 Gagarina Ave., Nizhny Novgorod, Russia, 603950

    D. B. Mokeev

  2. Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics, 136 Rodionova Str., Nizhny Novgorod, Russia, 603093

    D. B. Mokeev

  3. National Research University Higher School of Economics, 25/12 Bolshaja Pecherskaja Ulitsa, Nizhny Novgorod, Russia, 603155

    D. S. Malyshev

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  1. D. B. Mokeev
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  2. D. S. Malyshev
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Correspondence to D. S. Malyshev.

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Mokeev, D.B., Malyshev, D.S. A polynomial-time algorithm of finding a minimum k-path vertex cover and a maximum k-path packing in some graphs. Optim Lett 14, 1317–1322 (2020). https://doi.org/10.1007/s11590-019-01475-0

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  • DOI: https://doi.org/10.1007/s11590-019-01475-0

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