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On the Convexity Number of Graphs

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Abstract

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k. Furthermore, we identify natural necessary extension properties of graphs of small convexity number and study the interplay between these properties and upper bounds on the convexity number.

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil

    Mitre C. Dourado

  2. Instituto de Computação, Universidade Federal Fluminense, Niterói, RJ, Brazil

    Fábio Protti

  3. Institut für Optimierung und Operations Research, Universität Ulm, 89069, Ulm, Germany

    Dieter Rautenbach

  4. Instituto de Matemática, NCE and COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil

    Jayme L. Szwarcfiter

Authors
  1. Mitre C. Dourado
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  2. Fábio Protti
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  3. Dieter Rautenbach
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  4. Jayme L. Szwarcfiter
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Correspondence to Mitre C. Dourado.

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Dourado, M.C., Protti, F., Rautenbach, D. et al. On the Convexity Number of Graphs. Graphs and Combinatorics 28, 333–345 (2012). https://doi.org/10.1007/s00373-011-1049-7

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  • DOI: https://doi.org/10.1007/s00373-011-1049-7

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