Can a graph be drawn on a plane without any lines crossing? This paper introduces an efficient algorithm for planarity testing, determining whether an arbitrary graph can be embedded in a plane. This work builds on and improves upon a method proposed by Auslander and Parter and later refined by Goldstein. The algorithm leverages depth-first search, achieving *O*(*V*) time and space bounds, where *V* represents the number of vertices in the graph. A successful ALGOL implementation tested graphs with up to 900 vertices in under 12 seconds. This algorithm has significant applications in various fields, such as circuit board design and network visualization, where efficient planarity testing is crucial. The work could lead to faster and simpler techniques for determining whether a set of objects can be arranged to be non-overlapping, which can speed up processing in a computer and reduce its complexity.
This article was published in the Journal of the ACM. The paper contributes to the journal's coverage of computer science and engineering, particularly in the areas of computer hardware and software. By presenting an efficient algorithm for planarity testing, the research aligns with the journal's mission of advancing knowledge in computing and related fields.