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The Strong Convexity Spectra of Grids

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Abstract

Let D be a connected oriented graph. A set \(S \subseteq V(D)\) is convex in D if, for every pair of vertices \(x, y \in S\), the vertex set of every xy-geodesic, (xy shortest directed path) and every yx-geodesic in D is contained in S. The convexity number, \(\hbox {con}(D)\), of a non-trivial oriented graph, D, is the maximum cardinality of a proper convex set of D. The strong convexity spectrum of the graph G, \(S_{SC} (G)\), is the set \(\{\hbox {con}(D) :\ D \hbox { is a strong orientation of G} \}\). In this paper we prove that the problem of determining the convexity number of an oriented graph is \(\mathcal {NP}\)-complete, even for bipartite oriented graphs of arbitrary large girth, extending previous known results for graphs. We also determine \(S_{SC} (P_n \Box P_m)\), for every pair of integers \(n,m \ge 2\).

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Authors and Affiliations

  1. Universidad Nacional Autónoma de México, Campus Juriquilla, 76230, Juriquilla, Querétaro, México

    Gabriela Araujo-Pardo

  2. Universidad Nacional Autónoma de México, Área de la Investigación Científica, Circuito Exterior, C.U., C.P. 4510, Coyoacán, México CDMX, México

    César Hernández-Cruz & Juan José Montellano-Ballesteros

Authors
  1. Gabriela Araujo-Pardo
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  2. César Hernández-Cruz
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  3. Juan José Montellano-Ballesteros
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Corresponding author

Correspondence to César Hernández-Cruz.

Additional information

This research was supported by CONACyT-México under Projects 178395 and 166306 and PAPIIT-México under Projects IN101912 and IN104915.

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Araujo-Pardo, G., Hernández-Cruz, C. & Montellano-Ballesteros, J.J. The Strong Convexity Spectra of Grids. Graphs and Combinatorics 33, 689–713 (2017). https://doi.org/10.1007/s00373-017-1805-4

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  • DOI: https://doi.org/10.1007/s00373-017-1805-4

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