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Improved Bounds for Guarding Plane Graphs with Edges

Authors Ahmad Biniaz, Prosenjit Bose, Aurélien Ooms, Sander Verdonschot

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Author Details

Ahmad Biniaz
  • Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Aurélien Ooms
  • Département d'Informatique, Université libre de Bruxelles (ULB), Brussels, Belgium
Sander Verdonschot
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

  • Biniaz, Ahmad: Supported by NSERC and Fields postdoctoral fellowships.
  • Bose, Prosenjit: Supported by NSERC.
  • Ooms, Aurélien: Supported by the Fund for Research Training in Industry and Agriculture (FRIA).
  • Verdonschot, Sander: Partially supported by NSERC and the Carleton-Fields Postdoctoral Award.

Cite As Get BibTex

Ahmad Biniaz, Prosenjit Bose, Aurélien Ooms, and Sander Verdonschot. Improved Bounds for Guarding Plane Graphs with Edges. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.SWAT.2018.14

Abstract

An edge guard set of a plane graph G is a subset Gamma of edges of G such that each face of G is incident to an endpoint of an edge in Gamma. Such a set is said to guard G. 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 (1997) that G can be guarded with at most 2n/5 edges, then extend this approach with a deeper analysis to yield an improved bound of 3n/8 edges for any plane graph.
2) We prove that there exists an edge guard set of G with at most n/(3) + alpha/9 edges, where alpha is the number of quadrilateral faces in G. This improves the previous bound of n/(3) + alpha by Bose, Kirkpatrick, and Li (2003). Moreover, if there is no short path between any two quadrilateral faces in G, we show that n/(3) edges suffice, removing the dependence on alpha.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph theory
Keywords
  • edge guards
  • graph coloring
  • four-color theorem

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References

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