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- W4302177579 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 $k leq 4$ and NP-complete for $k geq 5$. 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 $k geq 7$ and even $k geq 10$. 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." @default.
- W4302177579 created "2022-10-06" @default.
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- W4302177579 date "2014-09-15" @default.
- W4302177579 modified "2023-10-18" @default.
- W4302177579 title "Planar Embeddings with Small and Uniform Faces" @default.
- W4302177579 doi "https://doi.org/10.48550/arxiv.1409.4299" @default.
- W4302177579 hasPublicationYear "2014" @default.
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