Plane 3-Trees: Embeddability and Approximation

Abstract

We give an $O(n\log ^3 n)$-time linear-space algorithm that, given a plane 3-tree $G$ with $n$ vertices and a set $S$ of $n$ points in the plane, determines whether $G$ has a point-set embedding on $S$ (i.e., a planar straight-line drawing of $G$ where each vertex is mapped to a distinct point of $S$), improving the $O(n^{4/3+\varepsilon})$-time $O(n^{4/3})$-space algorithm of Moosa and Rahman [Lecture Notes in Comput. Sci. 6842, Springer, New York, 2011, pp. 204--212]. Given an arbitrary plane graph $G$ and a point set $S$, Kaufmann and Wiese [J. Graph Algorithms Appl., 6 (2002), pp. 115--129] gave an algorithm to compute 2-bend point-set embeddings of $G$ on $S$. Later, Di Giacomo and Liotta [Lecture Notes in Comput. Sci. 5942, Springer, New York, 2010, pp. 35--46] showed how such a drawing can be computed using $O(W^3)$ area, where $W$ is the length of the longest edge of the bounding box of $S$. Their algorithm uses $O(W^3)$ area even when the input graphs are restricted to plane 3-trees. We introduce new techniques for computing $2$-bend point-set embeddings of plane 3-trees that take only $O(W^2)$ area. We also give approximation algorithms for point-set embeddings of plane $3$-trees. Our results on 2-bend point-set embeddings and approximate point-set embeddings hold for partial plane $3$-trees (e.g., series-parallel graphs and Halin graphs).