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Improved Bounds for Guarding Plane Graphs with Edges

  • Ahmad Biniaz
  • Prosenjit Bose
  • Aurélien Ooms
  • Sander Verdonschot
Original Paper
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Abstract

An edge guard set of a plane graph G is a subset \(\varGamma \) of edges of G such that each face of G is incident to an endpoint of an edge in \(\varGamma \). Such a set is said to guardG. We improve the known upper bounds on the number of edges required to guard any n-vertex embedded planar graph G: (1) We present a simple inductive proof for a theorem of Everett and Rivera-Campo (Comput Geom Theory Appl 7:201–203, 1997) that G can be guarded with at most \(\frac{2n}{5}\) edges, then extend this approach with a deeper analysis to yield an improved bound of \(\frac{3n}{8}\) edges for any plane graph. (2) We prove that there exists an edge guard set of G with at most \(\frac{n}{3} + \frac{\alpha }{9}\) edges, where \(\alpha \) is the number of quadrilateral faces in G. This improves the previous bound of \(\frac{n}{3} + \alpha \) by Bose et al. (Comput Geom Theory Appl 26(3):209–219, 2003). Moreover, if there is no short path between any two quadrilateral faces in G, we show that \(\frac{n}{3}\) edges suffice, removing the dependence on \(\alpha \).

Keywords

Edge guards Graph coloring Four-color theorem 

Notes

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  • Ahmad Biniaz
    • 1
  • Prosenjit Bose
    • 2
  • Aurélien Ooms
    • 3
  • Sander Verdonschot
    • 2
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Département d’InformatiqueUniversité libre de Bruxelles (ULB)BrusselsBelgium

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