COCOON 2009: Computing and Combinatorics pp 192-204

# Convex Partitions with 2-Edge Connected Dual Graphs

• Marwan Al-Jubeh
• Michael Hoffmann
• Mashhood Ishaque
• Diane L. Souvaine
• Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5609)

## 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 a 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.

## Preview

### References

1. 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.: Compatible geometric matchings. Comput. Geom. 42, 617–626 (2009)
2. 2.
Andrzejak, A., Aronov, B., Har-Peled, S., Seidel, R., Welzl, E.: Results on k-sets and j-facets via continuous motion. In: Proc. 14th SCG, pp. 192–199. ACM, New York (1998)Google Scholar
3. 3.
Banchoff, T.F.: Global geometry of polygons I: The theorem of Fabricius-Bjerre. Proc. AMS 45, 237–241 (1974)
4. 4.
Benbernou, N., Demaine, E.D., Demaine, M.L., Hoffmann, M., Ishaque, M., Souvaine, D.L., Tóth, C.D.: Disjoint segments have convex partitions with 2-edge connected dual graphs. In: Proc. CCCG, pp. 13–16 (2007)Google Scholar
5. 5.
Benbernou, N., Demaine, E.D., Demaine, M.L., Hoffmann, M., Ishaque, M., Souvaine, D.L., Tóth, C.D.: Erratum for Disjoint segments have convex partitions with 2-edge connected dual graphs. In: Proc. CCCG, p. 223 (2008)Google Scholar
6. 6.
Bose, P., Houle, M.E., Toussaint, G.T.: Every set of disjoint line segments admits a binary tree. Discrete Comput. Geom. 26(3), 387–410 (2001)
7. 7.
Carlsson, J.G., Armbruster, B., Ye, Y.: Finding equitable convex partitions of points in a polygon efficiently. ACM Transactions on Algorithms (to appear, 2009)Google Scholar
8. 8.
Chazelle, B., Dobkin, D.P.: Optimal Convex Decompositions. Comput. Geom. 2, 63–133 (1985)
9. 9.
Kaneko, A., Kano, M.: Perfect partitions of convex sets in the plane. Discrete Comput. Geom. 28(2), 211–222 (2002)
10. 10.
Keil, M.: Polygon decomposition. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 491–518. Elsevier, Amsterdam (2000)
11. 11.
Keil, M., Snoeyink, J.: On the time bound for convex decomposition of simple polygons. Internat. J. Comput. Geom. Appl. 12, 181–192 (2002)
12. 12.
Krumme, D.W., Rafalin, E., Souvaine, D.L., Tóth, C.D.: Tight bounds for connecting sites across barriers. Discrete Comput. Geom. 40(3), 377–394 (2008)
13. 13.
Lien, J.-M., Amato, N.M.: Approximate convex decomposition of polygons. Comput. Geom. 35(1-2), 100–123 (2006)
14. 14.
Lingas, A.: The power of non-rectilinear holes. In: Nielsen, M., Schmidt, E.M. (eds.) ICALP 1982. LNCS, vol. 140, pp. 369–383. Springer, Heidelberg (1982)

## 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 ZürichSwitzerland
3. 3.Department of MathematicsUniversity of CalgaryCanada