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 3and 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.