Packing seagulls

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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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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Mathematics Subject Classification (2000)

  • 05C83
  • 05C70