On the Kőnig-Egerváry theorem for $k$-paths - LIRMM - Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier Access content directly
Journal Articles Journal of Graph Theory Year : 2019

On the Kőnig-Egerváry theorem for $k$-paths

Abstract

The famous Kőnig‐Egerváry theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we consider the set Gk of all graphs such that, for every induced subgraph, the maximum number of disjoint paths of order k equals the minimum order of a set of vertices intersecting all paths of order k. For k ∈ {3,4}, we give complete structural descriptions of the graphs in Gk. Furthermore, for odd k, we give a complete structural description of the graphs in Gk that contain no cycle of order less than k. For these graph classes, our results yield efficient recognition algorithms as well as efficient algorithms that determine maximum sets of disjoint paths of order k and minimum sets of vertices intersecting all paths of order k.

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Dates and versions

lirmm-02078255 , version 1 (25-03-2019)

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Stéphane Bessy, Pascal Ochem, Dieter Rautenbach. On the Kőnig-Egerváry theorem for $k$-paths. Journal of Graph Theory, 2019, 91 (1), pp.73-87. ⟨10.1002/jgt.22421⟩. ⟨lirmm-02078255⟩
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