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The Sandwich Problem for Decompositions and Almost Monotone Properties

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Abstract

We consider the graph sandwich problem and introduce almost monotone properties, for which the sandwich problem can be reduced to the recognition problem. We show that the property of containing a graph in \({\mathcal {C}}\) as an induced subgraph is almost monotone if \({\mathcal {C}}\) is the set of thetas, the set of pyramids, or the set of prisms and thetas. We show that the property of containing a hole of length \(\equiv j \mod n\) is almost monotone if and only if \(j \equiv 2 \mod n\) or \(n \le 2\). Moreover, we show that the imperfect graph sandwich problem, also known as the Berge trigraph recognition problem, can be solved in polynomial time. We also study the complexity of several graph decompositions related to perfect graphs, namely clique cutset, (full) star cutset, homogeneous set, homogeneous pair, and 1-join, with respect to the partitioned and unpartitioned probe problems. We show that the clique cutset and full star cutset unpartitioned probe problems are NP-hard. We show that for these five decompositions, the partitioned probe problem is in P, and the homogeneous set, 1-join, 1-join in the complement, and full star cutset in the complement unpartitioned probe problems can be solved in polynomial time as well.

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Acknowledgements

We are thankful to Paul Seymour for many helpful discussions. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under Grant No. W911NF-16-1-0404.

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Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08544, USA

    Maria Chudnovsky & Sophie Spirkl

  2. COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Celina M. H. de Figueiredo

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  1. Maria Chudnovsky
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  2. Celina M. H. de Figueiredo
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  3. Sophie Spirkl
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Correspondence to Sophie Spirkl.

Additional information

Maria Chudnovsky was supported by National Science Foundation Grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404. Celina M. H. de Figueiredo was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq Grant 303622/2011-3.

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Chudnovsky, M., de Figueiredo, C.M.H. & Spirkl, S. The Sandwich Problem for Decompositions and Almost Monotone Properties. Algorithmica 80, 3618–3645 (2018). https://doi.org/10.1007/s00453-018-0409-6

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  • DOI: https://doi.org/10.1007/s00453-018-0409-6

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