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Listing All Fixed-Length Simple Cycles in Sparse Graphs in Optimal Time

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Abstract

The degeneracy of an n-vertex graph G is the smallest number k such that every subgraph of G contains a vertex of degree at most k. We present an algorithm for enumerating all simple cycles of length p in an n-order k-degenerate graph running in time \(\mathcal {O}(n^{\lfloor {p/2} \rfloor } k^{\lceil p/2 \rceil })\). We then show that this algorithm is worst-case output size optimal by proving a \(\varTheta (n^{\lfloor {p/2} \rfloor } k^{\lceil {p}/{2} \rceil })\) bound on the maximal number of simple p-length cycles in these graphs. Our results also apply to induced (chordless) cycles.

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Authors and Affiliations

  1. LRI, CNRS, Université Paris Sud, Université Paris Saclay, 91405, Orsay, France

    George Manoussakis

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  1. George Manoussakis
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Correspondence to George Manoussakis .

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Editors and Affiliations

  1. CNRS, University of Bordeaux, Talence cedex, France

    Ralf Klasing

  2. LaBRI, University of Bordeaux, Talence cedex, France

    Marc Zeitoun

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Manoussakis, G. (2017). Listing All Fixed-Length Simple Cycles in Sparse Graphs in Optimal Time. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_28

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  • DOI: https://doi.org/10.1007/978-3-662-55751-8_28

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  • Online ISBN: 978-3-662-55751-8

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