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

1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
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.
Keywords :
Document type :
Journal articles
Domain :
Complete list of metadata

Cited literature [13 references]

https://hal-lirmm.ccsd.cnrs.fr/lirmm-02078255
Contributor : Alexandre Pinlou Connect in order to contact the contributor
Submitted on : Monday, March 25, 2019 - 10:53:17 AM
Last modification on : Wednesday, November 4, 2020 - 6:08:04 PM
Long-term archiving on: : Wednesday, June 26, 2019 - 1:25:44 PM

### Citation

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

Record views