Finding minimum clique capacity


Let C be a clique of a graph G. The capacity of C is defined to be (|V (G)\C|+|D|)/2, where D is the set of vertices in V (G)\C that have both a neighbour and a non-neighbour in C. We give a polynomial-time algorithm to find the minimum clique capacity in a graph G. This problem arose in the study [1] of packing vertex-disjoint induced three-vertex paths in a graph with no stable set of size three, which in turn was motivated by Hadwiger’s conjecture.

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  1. [1]

    M. Chudnovsky and P. Seymour: Packing seagulls, Combinatorica, this issue.

  2. [2]

    J. E. Hopcroft and R. M. Karp: An n 5/2 algorithm for maximum matchings in bipartite graphs, SIAM Journal on Computing 2 (1973), 225–231.

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Corresponding author

Correspondence to Maria Chudnovsky.

Additional information

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

Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0011653) and a TJ Park Junior Faculty Fellowship.

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

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Chudnovsky, M., Oum, SI. & Seymour, P. Finding minimum clique capacity. Combinatorica 32, 283–287 (2012).

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Mathematics Subject Classification (2000)

  • 07C70
  • 05C85
  • 05C69