Graphs and Combinatorics

, Volume 35, Issue 2, pp 437–450 | Cite as

Improved Bounds for Guarding Plane Graphs with Edges

  • Ahmad BiniazEmail author
  • Prosenjit Bose
  • Aurélien Ooms
  • Sander Verdonschot
Original Paper


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 \(\alpha \).


Edge guards Graph coloring Four-color theorem 



  1. 1.
    Appel, K., Haken, W.: Every Planar Map is Four Colorable. Contemporary Mathematics, vol. 98. American Mathematical Society, Providence (1989). With the collaboration of J. KochzbMATHGoogle Scholar
  2. 2.
    Borodin, O.V.: Structure of neighborhoods of an edge in planar graphs and the simultaneous coloring of vertices, edges, and faces. Matematicheskie Zametki 53(5), 35–47 (1993)MathSciNetGoogle Scholar
  3. 3.
    Bose, P., Kirkpatrick, D.G., Li, Z.: Worst-case-optimal algorithms for guarding planar graphs and polyhedral surfaces. Comput. Geom. Theory Appl. 26(3), 209–219 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bose, P., Shermer, T.C., Toussaint, G.T., Zhu, B.: Guarding polyhedral terrains. Comput. Geom. Theory Appl. 7, 173–185 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chvátal, V.: A combinatorial theorem in plane geometry. J. Comb. Theory Ser. B 18, 39–41 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Everett, H., Rivera-Campo, E.: Edge guarding polyhedral terrains. Comput. Geom. Theory Appl. 7, 201–203 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fisk, S.: A short proof of Chvatal’s watchman theorem. J. Comb. Theory Ser. B 24(3), 374 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lebesgue, H.: Quelques conséquences simple de la formula d’Euler. Journal de Mathématiques Pures et Appliquées 19, 27–43 (1940)zbMATHGoogle Scholar
  9. 9.
    O’Rourke, J.: Galleries need fewer mobile guards: a variation on Chvatal’s theorem. Geometriae Dedicata 14, 273–283 (1983)MathSciNetzbMATHGoogle Scholar
  10. 10.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press, Oxford (1987)zbMATHGoogle Scholar
  11. 11.
    Sack, J., Urrutia, J. (eds.): Handbook of Computational Geometry. North-Holland, Amsterdam (2000)zbMATHGoogle Scholar
  12. 12.
    Shermer, T.C.: Recent results in art galleries. Proc. IEEE 80, 1384–1399 (1992)CrossRefGoogle Scholar
  13. 13.
    Toth, C.D., O’Rourke, J., Goodman, J.E. (eds.): Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (2017)zbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  • Ahmad Biniaz
    • 1
    Email author
  • 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

Personalised recommendations