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Rooted Cycle Bases

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Algorithms and Data Structures (WADS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9214))

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Abstract

A cycle basis in an undirected graph is a minimal set of simple cycles whose symmetric differences include all Eulerian subgraphs of the given graph. We define a rooted cycle basis to be a cycle basis in which all cycles contain a specified root edge, and we investigate the algorithmic problem of constructing rooted cycle bases. We show that a given graph has a rooted cycle basis if and only if the root edge belongs to its 2-core and the 2-core is 2-vertex-connected, and that constructing such a basis can be performed efficiently. We show that in an unweighted or positively weighted graph, it is possible to find the minimum weight rooted cycle basis in polynomial time. Additionally, we show that it is \(\mathsf {NP}\)-complete to find a fundamental rooted cycle basis (a rooted cycle basis in which each cycle is formed by combining paths in a fixed spanning tree with a single additional edge) but that the problem can be solved by a fixed-parameter-tractable algorithm when parameterized by clique-width.

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Author information

Authors and Affiliations

  1. Computer Science Department, University of California, Irvine, USA

    David Eppstein

  2. Department of Mechanical and Aerospace Engineering, University of California, Irvine, USA

    J. Michael McCarthy & Brian E. Parrish

Authors
  1. David Eppstein
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  2. J. Michael McCarthy
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  3. Brian E. Parrish
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Corresponding author

Correspondence to David Eppstein .

Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. Carleton University, Ottawa, Canada

    Jörg-Rüdiger Sack

  3. University of Victoria, Victoria, Canada

    Ulrike Stege

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Eppstein, D., McCarthy, J.M., Parrish, B.E. (2015). Rooted Cycle Bases. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_28

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  • DOI: https://doi.org/10.1007/978-3-319-21840-3_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21839-7

  • Online ISBN: 978-3-319-21840-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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