Stop Minding Your P’s and Q’s: Implementing a Fast and Simple DFS-Based Planarity Testing and Embedding Algorithm

Abstract

Abstract. In this paper we give a new description of the planarity testing and embedding algorithm presented by Boyer and Myrvold [2], providing, in our opinion, new insights on the combinatorial foundations of the algorithm. Especially, we give a detailed illustration of a fundamental phase of the algorithm, called walk-up, which was only succinctly illustrated in [2]. Also, we present an implementation of the algorithm and extensively test its efficiency against the most popular implementations of planarity testing algorithms. Further, as a side effect of the test activity, we propose a general overview of the state of the art (restricted to efficiency issues) of the planarity testing and embedding field.

Additional PC-Tree Planarity Conditions

“…While the August 2003 version of the implementation in [10] could not be empirically compared due to frequent incorrect results, Hsu also requested that a subsequent version with fixes not be empirically compared as he felt the implementation was only a proof of concept. However, there is strong evidence from [11] that a simplified vertex addition method can achieve far better performance than most prior methods, although those results also suggest that the edge addition methods in [4,12] are faster. Future work must use the results of this paper to create correct, efficient PC-tree implementations for empirical comparisons, especially with [4,12].…”

“…Specifically, suppose the closest common ancestor m of the two terminal nodes is a C-node whose parent has the only child i * -subtree along the path P . Figure 3 depicts an example PC-tree and the corresponding K 5 minor pattern from [3]. In this case, the K 3,3 shown in the proof of Lemma 2.5 in [5] cannot be found, illustrating that the proof does not "go through for the case of general trees without any changes provided that the paths through a C-node are interpreted correctly" [5, p. 185].…”

“…It does not result in three terminal nodes as stated in [5], but is instead discovered by violation of the planarity condition in Theorem 4. The corresponding K5 minor from [3]. Note: Due to the difference in definitions between graph minors and subgraph homeomorphism, this case implies a subgraph homeomorphic to K3,3 or K5.…”

“…Recent research has produced simpler linear-time planarity algorithms [3][4][5]. This paper discusses the planarity method of Shih and Hsu [5], which is based on a data structure called a PC-tree.…”

A Note on the Self-similarity of Some Orthogonal Drawings

“…Each edge is guaranteed to have a single bend. TSM: For the topology-shape-metrics approach we used the Boyer and Myrvold algorithm [1] to compute a planar embedding. During the subsequent orthogonalization step, the total number of bends was minimized by using the algorithm in [10], modified as described in [5] for handling high degree graphs.…”