Planar Embeddings with Small and Uniform Faces

Abstract

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 $k \le 4$ and NP-complete for $k \ge 5$. 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 $k \ge 7$ and even $k \ge 10$. 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.

Work by Giordano Da Lozzo was supported in part by the Italian Ministry of Education, University, and Research (MIUR) under PRIN 2012C4E3KT national research project “AMANDA – Algorithmics for MAssive and Networked DAta”. Work by Jan Kratochvíl and Vít Jelínek was supported by the grant no. 14-14179S of the Czech Science Foundation GAČR. Ignaz Rutter was supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD).