Journal of Combinatorial Optimization

, Volume 22, Issue 3, pp 409–425 | Cite as

Convex partitions with 2-edge connected dual graphs

  • Marwan Al-Jubeh
  • Michael Hoffmann
  • Mashhood Ishaque
  • Diane L. Souvaine
  • Csaba D. Tóth
Article

Abstract

It is shown that for every finite set of disjoint convex polygonal obstacles in the plane, with a total of n vertices, the free space around the obstacles can be partitioned into open convex cells whose dual graph (defined below) is 2-edge connected. Intuitively, every edge of the dual graph corresponds to a pair of adjacent cells that are both incident to the same vertex.

Aichholzer et al. recently conjectured that given an even number of line-segment obstacles, one can construct a convex partition by successively extending the segments along their supporting lines such that the dual graph is the union of two edge-disjoint spanning trees. Here we present counterexamples to this conjecture, with n disjoint line segments for any n≥15, such that the dual graph of any convex partition constructed by this method has a bridge edge, and thus the dual graph cannot be partitioned into two spanning trees.

Keywords

Convex partitions Dual graphs Geometric matchings 

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References

  1. Aichholzer O, Bereg S, Dumitrescu A, García A, Huemer C, Hurtado F, Kano M, Márquez A, Rappaport D, Smorodinsky S, Souvaine D, Urrutia J, Wood D (2009) Compatible geometric matchings. Comput Geom 42:617–626 MathSciNetMATHCrossRefGoogle Scholar
  2. Andrzejak A, Aronov B, Har-Peled S, Seidel R, Welzl E (1998) Results on k-sets and j-facets via continuous motion. In: Proc 14th Sympos Comput Geom. ACM, New York, pp 192–199 Google Scholar
  3. Banchoff TF (1974) Global geometry of polygons I: The theorem of Fabricius–Bjerre. Proc Am Math Soc 45:237–241 MathSciNetMATHCrossRefGoogle Scholar
  4. Benbernou N, Demaine ED, Demaine ML, Hoffmann M, Ishaque M, Souvaine DL, Tóth CD (2007) Disjoint segments have convex partitions with 2-edge connected dual graphs. In: Proc Canadian Conf Comput Geom, pp 13–16 Google Scholar
  5. Benbernou N, Demaine ED, Demaine ML, Hoffmann M, Ishaque M, Souvaine DL, Tóth CD (2008) Erratum for “Disjoint segments have convex partitions with 2-edge connected dual graphs”. In: Proc Canadian Conf Comput Geom, p 223 Google Scholar
  6. Bentley JL, Ottmann TA (1979) Algorithms for reporting and counting geometric intersections. IEEE Trans Comput C 28(9):643–647 MATHCrossRefGoogle Scholar
  7. Bose P, Houle ME, Toussaint GT (2001) Every set of disjoint line segments admits a binary tree. Discrete Comput Geom 26(3):387–410 MathSciNetMATHGoogle Scholar
  8. Carlsson JG, Armbruster B, Ye Y (2009) Finding equitable convex partitions of points in a polygon efficiently. ACM Trans Algorithms (to appear) Google Scholar
  9. Chazelle B, Dobkin DP (1985) Optimal convex decompositions. Comput Geom 2:63–133 MathSciNetGoogle Scholar
  10. Grantson M, Levcopoulos C (2005) A fixed parameter algorithm for the minimum number convex partition problem. In: Proc Japan Conf Discrete Comput Geom, Tokyo, 2004. LNCS, vol 3746. Springer, Berlin, pp 83–94 CrossRefGoogle Scholar
  11. Ishaque M, Speckmann B, Tóth CD (2009) Shooting permanent rays among disjoint polygons in the plane. In: Proc 25th Sympos Comput Geom. ACM, New York, pp 51–60 Google Scholar
  12. Kaneko A, Kano M (2002) Perfect partitions of convex sets in the plane. Discrete Comput Geom 28(2):211–222 MathSciNetMATHGoogle Scholar
  13. Keil M (2000) Polygon decomposition. In: Sack J-R, Urrutia J (eds) Handbook of computational geometry. Elsevier, Amsterdam, pp 491–518 CrossRefGoogle Scholar
  14. Keil M, Snoeyink J (2002) On the time bound for convex decomposition of simple polygons. Int J Comput Geom Appl 12:181–192 MathSciNetMATHCrossRefGoogle Scholar
  15. Knauer C, Spillner A (2006) Approximation algorithms for the minimum convex partition problem. In: Proc SWAT. LNCS, vol 4059. Springer, Berlin, pp 232–241 Google Scholar
  16. Krumme DW, Rafalin E, Souvaine DL, Tóth CD (2008) Tight bounds for connecting sites across barriers. Discrete Comput Geom 40(3):377–394 MathSciNetMATHCrossRefGoogle Scholar
  17. Lien J-M, Amato NM (2006) Approximate convex decomposition of polygons. Comput Geom 35(1–2):100–123 MathSciNetMATHCrossRefGoogle Scholar
  18. Lingas A (1982) The power of non-rectilinear holes. In: Proc 9th ICALP. LNCS, vol 140. Springer, Berlin, pp 369–383 Google Scholar
  19. Streinu I (2005) Pseudo-triangulations, rigidity and motion planning. Discrete Comput Geom 34(4):587–635 MathSciNetMATHCrossRefGoogle Scholar
  20. Tan G, Bertier M, Kermarrec A-M (2009) Convex partition of sensor networks and its use in virtual coordinate geographic routing. In: Proc INFOCOM. IEEE Comput Soc, Los Alamitos, pp 1746–1754 Google Scholar
  21. Tóth CD (2003) Guarding disjoint triangles and claws in the plane. Comput Geom 25(1–2):51–65 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Marwan Al-Jubeh
    • 1
  • Michael Hoffmann
    • 2
  • Mashhood Ishaque
    • 1
  • Diane L. Souvaine
    • 1
  • Csaba D. Tóth
    • 3
  1. 1.Department of Computer ScienceTufts UniversityMefordUSA
  2. 2.Institute of Theoretical Computer ScienceETH ZürichZurichSwitzerland
  3. 3.Department of MathematicsUniversity of CalgaryCalgaryCanada

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