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Packing seagulls

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Abstract

A seagull in a graph is an induced three-vertex path. When does a graph G have k pairwise vertex-disjoint seagulls? This is NP-complete in general, but for graphs with no stable set of size three we give a complete solution. This case is of special interest because of a connection with Hadwiger’s conjecture which was the motivation for this research; and we deduce a unification and strengthening of two theorems of Blasiak [2] concerned with Hadwiger’s conjecture.

Our main result is that a graph G (different from the five-wheel) with no three-vertex stable set contains k disjoint seagulls if and only if

  1. •

    |V (G)|≥3k

  2. •

    G is k-connected

  3. •

    for every clique C of G, if D denotes the set of vertices in V (G)\C that have both a neighbour and a non-neighbour in C then |D|+|V (G)\C|≥2k, and

  4. •

    the complement graph of G has a matching with k edges.

We also address the analogous fractional and half-integral packing questions, and give a polynomial time algorithm to test whether there are k disjoint seagulls.

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References

  1. C. Berge: Sur le couplage maximum d’un graphe, C.R. Acad. Sci. Paris 247 (1958), 258–259.

    MathSciNet  MATH  Google Scholar 

  2. J. Blasiak: A special case of Hadwiger’s conjecture, J. Combinatorial Theory, Ser. B, 97 (2007), 1056–1073.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Chudnovsky, S.-I. Oum and P. Seymour: Finding minimum clique capacity, Combinatorics this issue.

  4. D. Dor and M. Tarsi: Graph decomposition is NP-complete: A complete proof of Holyer’s conjecture, SIAM J. Comput. 26 (1997), 1166–1187.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Edmonds: Matroid partition, in: Mathematics of the Decision Sciences Part 1 (Proceedings Fifth Summer Seminar on the Mathematics of the Decision Sciences, Stanford, California, 1967; G. B. Dantzig, A. F. Vienott, Jr. eds.) [Lectures in Applied Mathematics Vol.11], American Mathematical Society, Providence, Rhode Island, 1968, 333–345.

    Google Scholar 

  6. H. Hadwiger: Über eine Klassifikation der Streckenkomplexe, Vierteljahrsschr. Naturforsch. Ges. Zürich 88 (1943), 133–142.

    MathSciNet  Google Scholar 

  7. M. D. Plummer, M. Stiebitz and B. Toft: On a special case of Hadwiger’s conjecture, Discuss. Math. Graph Theory 23 (2005), 333–363.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Columbia University, New York, NY, 10027, USA

    Maria Chudnovsky

  2. Princeton University, Princeton, NJ, 08540, USA

    Paul Seymour

Authors
  1. Maria Chudnovsky
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  2. Paul Seymour
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Corresponding author

Correspondence to Maria Chudnovsky.

Additional information

Supported by NSF grants DMS-1001091 and IIS-1117631.

Supported by ONR grant N00014-10-1-0680 and NSF grant DMS-0901075.

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Chudnovsky, M., Seymour, P. Packing seagulls. Combinatorica 32, 251–282 (2012). https://doi.org/10.1007/s00493-012-2594-2

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  • DOI: https://doi.org/10.1007/s00493-012-2594-2

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