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On Distance-3 Matchings and Induced Matchings

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Graph Theory, Computational Intelligence and Thought

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5420))

Abstract

For a finite undirected graph G = (V,E) and positive integer k ≥ 1, an edge set M ⊆ E is a distance-k matching if the mutual distance of edges in M is at least k in G. For k = 1, this gives the usual notion of matching in graphs, and for general k ≥ 1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k = 2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.

Finding a maximum induced matching is \(\mathbb{NP}\)-complete even on very restricted bipartite graphs but for k = 2, it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)2 of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.

We show that, unlike for k = 2, for a chordal graph G, L(G)3 is not necessarily chordal, and finding a maximum distance-3 matching remains \(\mathbb{NP}\)-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-3 matching problem can be solved in polynomial time. Moreover, we obtain various new results for induced matchings.

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Author information

Authors and Affiliations

  1. Institut für Informatik, Universität Rostock, D-18051, Rostock, Germany

    Andreas Brandstädt

  2. Dipartimento di Scienze, Universitá degli Studi “G. D’Annunzio”, Pescara, 65121, Italy

    Raffaele Mosca

Authors
  1. Andreas Brandstädt
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  2. Raffaele Mosca
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Editor information

Editors and Affiliations

  1. The Caesarea Rothschild Institute, Mount Carmel, University of Haifa, 31905, Haifa, Israel

    Marina Lipshteyn

  2. Department of Computer Science and Mathematics, Ariel University Center of Samaria, 40700, Ariel, Israel

    Vadim E. Levit

  3. Department of Computer Science, Colorado State University, 80523-1873, Fort Collins, CO, USA

    Ross M. McConnell

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Brandstädt, A., Mosca, R. (2009). On Distance-3 Matchings and Induced Matchings. In: Lipshteyn, M., Levit, V.E., McConnell, R.M. (eds) Graph Theory, Computational Intelligence and Thought. Lecture Notes in Computer Science, vol 5420. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02029-2_11

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  • DOI: https://doi.org/10.1007/978-3-642-02029-2_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02028-5

  • Online ISBN: 978-3-642-02029-2

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