The three-in-a-tree problem

Abstract

We show that there is a polynomial time algorithm that, given three vertices of a graph, tests whether there is an induced subgraph that is a tree, containing the three vertices. (Indeed, there is an explicit construction of the cases when there is no such tree.) As a consequence, we show that there is a polynomial time algorithm to test whether a graph contains a “theta” as an induced subgraph (this was an open question of interest) and an alternative way to test whether a graph contains a “pyramid” (a fundamental step in checking whether a graph is perfect).

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References

  1. [1]

    D. Bienstock: On the complexity of testing for even holes and induced odd paths, Discrete Math.90 (1991), 85–92. Corrigendum: 102 (1992), 109.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    M. Chudnovsky and R. Kapadia: Detecting a theta or a prism, SIAM J. Discrete Math.22 (2008), 1–18.

    Article  MathSciNet  Google Scholar 

  3. [3]

    M. Chudnovsky, G. Cornuéjols, X. Liu, P. Seymour and K. Vušković: Recognizing Berge graphs, Combinatorica25(2) (2005), 143–186.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    R. Kapadia: Detecting a Theta or a Prism, Senior Thesis, Princeton, May 2006.

  5. [5]

    F. Maffray and N. Trotignon: Algorithms for perfectly contractile graphs, SIAM J. Discrete Math.19 (2005), 553–574.

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Paul Seymour.

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This research was conducted while the author served as a Clay Mathematics Institute Research Fellow.

Supported by ONR grant N00014-01-1-0608, and NSF grant DMS-0070912.

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Chudnovsky, M., Seymour, P. The three-in-a-tree problem. Combinatorica 30, 387–417 (2010). https://doi.org/10.1007/s00493-010-2334-4

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Mathematics Subject Classification (2000)

  • 05C75