Maximum Matchings and Minimum Blocking Sets in \(\varTheta _6\)-Graphs

  • Therese Biedl
  • Ahmad Biniaz
  • Veronika Irvine
  • Kshitij Jain
  • Philipp KindermannEmail author
  • Anna Lubiw
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11789)


\(\varTheta _6\)-graphs are important geometric graphs that have many applications especially in wireless sensor networks. They are equivalent to Delaunay graphs where empty equilateral triangles take the place of empty circles. We investigate lower bounds on the size of maximum matchings in these graphs. The best known lower bound is n/3, where n is the number of vertices of the graph, which comes from half-\(\varTheta _6\)-graphs that are subgraphs of \(\varTheta _6\)-graphs. Babu et al. (2014) conjectured that any \(\varTheta _6\)-graph has a (near-)perfect matching (as is true for standard Delaunay graphs). Although this conjecture remains open, we improve the lower bound to \((3n-8)/7\).

We also relate the size of maximum matchings in \(\varTheta _6\)-graphs to the minimum size of a blocking set. Every edge of a \(\varTheta _6\)-graph on point set P corresponds to an empty triangle that contains the endpoints of the edge but no other point of P. A blocking set has at least one point in each such triangle. We prove that the size of a maximum matching is at least \(\beta (n)/2\) where \(\beta (n)\) is the minimum, over all \(\varTheta _6\)-graphs with n vertices, of the minimum size of a blocking set. In the other direction, lower bounds on matchings can be used to prove bounds on \(\beta \), allowing us to show that \(\beta (n)\ge 3n/4-2\).


Theta-six graphs Proximity graphs Maximum matching Minimum blocking set Triangular-distance Delaunay graph 



This work was done by a University of Waterloo problem solving group. We thank the other participants, Alexi Turcotte and Anurag Murty Naredla, for inspiring discussions, and the anonymous reviewers for helpful comments.


  1. 1.
    Ábrego, B.M., et al.: Matching points with circles and squares. In: Akiyama, J., Kano, M., Tan, X. (eds.) JCDCG 2004. LNCS, vol. 3742, pp. 1–15. Springer, Heidelberg (2005). Scholar
  2. 2.
    Ábrego, B.M., et al.: Matching points with squares. Discrete Comput. Geom. 41(1), 77–95 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aichholzer, O., et al.: Blocking Delaunay triangulations. Comput. Geom.: Theory Appl. 46(2), 154–159 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alzoubi, K.M., Li, X., Wang, Y., Wan, P., Frieder, O.: Geometric spanners for wireless ad hoc networks. IEEE Trans. Parallel Distrib. Syst. 14(4), 408–421 (2003)CrossRefGoogle Scholar
  5. 5.
    Aronov, B., Dulieu, M., Hurtado, F.: Witness (Delaunay) graphs. Comput. Geom.: Theory Appl. 44(6–7), 329–344 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Aronov, B., Dulieu, M., Hurtado, F.: Witness Gabriel graphs. Comput. Geom.: Theory Appl. 46(7), 894–908 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Aurenhammer, F., Paulini, G.: On shape Delaunay tessellations. Inf. Process. Lett. 114(10), 535–541 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Babu, J., Biniaz, A., Maheshwari, A., Smid, M.H.M.: Fixed-orientation equilateral triangle matching of point sets. Theor. Comput. Sci. 555, 55–70 (2014). Also in WALCOM 2013MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bauer, D., Broersma, H., Schmeichel, E.: Toughness in graphs–a survey. Graphs Comb. 22(1), 1–35 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Berge, C.: Sur le couplage maximum d’un graphe. Comptes Rendus de l’Académie des Sciences, Paris 247, 258–259 (1958)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Biedl, T., Biniaz, A., Irvine, V., Jain, K., Kindermann, P., Lubiw, A.: Maximum matchings and minimum blocking sets in \(\theta _6\)-graphs. Arxiv report (2019).
  12. 12.
    Biniaz, A., Maheshwari, A., Smid, M.H.M.: Higher-order triangular-distance Delaunay graphs: graph-theoretical properties. Comput. Geom.: Theory Appl. 48(9), 646–660 (2015). Also in CALDAM 2015MathSciNetCrossRefGoogle Scholar
  13. 13.
    Biniaz, A., Maheshwari, A., Smid, M.H.M.: Matchings in higher-order Gabriel graphs. Theor. Comput. Sci. 596, 67–78 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bonichon, N., Gavoille, C., Hanusse, N., Ilcinkas, D.: Connections between theta-graphs, delaunay triangulations, and orthogonal surfaces. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 266–278. Springer, Heidelberg (2010). Scholar
  15. 15.
    Bose, P., De Carufel, J.L., Hill, D., Smid, M.H.M.: On the spanning and routing ratio of theta-four. In: Chan, T.M. (ed.) Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2361–2370. SIAM (2019)CrossRefGoogle Scholar
  16. 16.
    Bose, P., Fagerberg, R., Van Renssen, A., Verdonschot, S.: Competitive routing in the half-\(\theta _6\)-graph. In: Rabani, Y. (ed.) Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1319–1328. SIAM (2012)Google Scholar
  17. 17.
    Bose, P., Morin, P., van Renssen, A., Verdonschot, S.: The \(\theta _5\)-graph is a spanner. Comput. Geom. 48(2), 108–119 (2015). Also in WG 2013MathSciNetCrossRefGoogle Scholar
  18. 18.
    Chew, P.: There are planar graphs almost as good as the complete graph. J. Comput. Syst. Sci. 39(2), 205–219 (1989)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Clarkson, K.L.: Approximation algorithms for shortest path motion planning. In: Aho, A.V. (ed.) Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), pp. 56–65. ACM (1987)Google Scholar
  20. 20.
    Damian, M., Iacono, J., Winslow, A.: Spanning properties of Theta-Theta-6. arXiv:1808.04744 (2018)
  21. 21.
    Dillencourt, M.B.: Toughness and Delaunay triangulations. Discrete Comput. Geom. 5, 575–601 (1990)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Drysdale III, R.L.S.: A practical algorithm for computing the Delaunay triangulation for convex distance functions. In: Proceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 159–168 (1990)Google Scholar
  23. 23.
    Fischer, M., Lukovszki, T., Ziegler, M.: Geometric searching in walkthrough animations with weak spanners in real time. In: Bilardi, G., Italiano, G.F., Pietracaprina, A., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 163–174. Springer, Heidelberg (1998). Scholar
  24. 24.
    Keil, J.M.: Approximating the complete Euclidean graph. In: Karlsson, R., Lingas, A. (eds.) SWAT 1988. LNCS, vol. 318, pp. 208–213. Springer, Heidelberg (1988). Scholar
  25. 25.
    Keil, J.M., Gutwin, C.A.: Classes of graphs which approximate the complete Euclidean graph. Discrete Comput. Geom. 7, 13–28 (1992)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Morin, P., Verdonschot, S.: On the average number of edges in Theta graphs. Online J. Anal. Comb., page to appear (2014). Also in ANALCO 2014Google Scholar
  27. 27.
    Nishizeki, T., Baybars, I.: Lower bounds on the cardinality of the maximum matchings of planar graphs. Discrete Math. 28(3), 255–267 (1979)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tutte, W.T.: The factorization of linear graphs. J. Lond. Math. Soc. 22, 107–111 (1947)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Therese Biedl
    • 1
  • Ahmad Biniaz
    • 1
  • Veronika Irvine
    • 1
  • Kshitij Jain
    • 2
  • Philipp Kindermann
    • 3
    Email author
  • Anna Lubiw
    • 1
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Borealis AIWaterlooCanada
  3. 3.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany

Personalised recommendations