Planar Embeddings with Small and Uniform Faces
Motivated by finding planar embeddings that lead to drawings with favorable aesthetics, we study the problems MINMAXFACE and UNIFORMFACES of embedding a given biconnected multi-graph such that the largest face is as small as possible and such that all faces have the same size, respectively.
We prove a complexity dichotomy for MINMAXFACE and show that deciding whether the maximum is at most k is polynomial-time solvable for k≤4 and NP-complete for k≥5. Further, we give a 6-approximation for minimizing the maximum face in a planar embedding. For UNIFORMFACES, we show that the problem is NP-complete for odd k≥7 and even k≥10. Moreover, we characterize the biconnected planar multi-graphs admitting 3– and 4-uniform embeddings (in a k-uniform embedding all faces have size k) and give an efficient algorithm for testing the existence of a 6-uniform embedding.
References:
- Giordano Da Lozzo, Vìt Jelìnek, Jan Kratochvìl, Ignaz Rutter. Planar Embeddings with Small and Uniform Faces. In, Hee-Kap Ahn, Chan-Su Shin, editors, Proc. 25th International Symposium on Algorithms and Computation (ISAAC 2014), Springer-Verlag, volume 8889 of Lecture Notes in Computer Science, pages 633-645, 2014.
- Giordano Da Lozzo, Vìt Jelìnek, Jan Kratochvìl, Ignaz Rutter. Planar Embeddings with Small and Uniform Faces. Technical Report arXiv:1409.4299, Cornell University, 2014.
