Planar Embeddings with Small and Uniform Faces

TL;DR

This paper investigates the computational complexity of creating planar graph embeddings with small or uniform face sizes, providing complexity classifications, approximation algorithms, and characterizations for specific cases.

Contribution

It establishes complexity dichotomies for MINMAXFACE and UNIFORMFACES problems, and offers algorithms and characterizations for certain uniform face embeddings.

Findings

Deciding max face size ≤ 4 is polynomial; ≥ 5 is NP-complete.

A 6-approximation algorithm for minimizing maximum face size.

NP-completeness results for odd and even uniform face size embeddings.

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.