Simplified O(n) Algorithms for Planar Graph Embedding, Kuratowski Subgraph Isolation, and Related Problems

Download or read book Simplified O(n) Algorithms for Planar Graph Embedding, Kuratowski Subgraph Isolation, and Related Problems written by John M. Boyer and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections. Due to the wealth of interest from the computer science community, there are a number of remarkable but complex O(n) planar embedding algorithms. This dissertation presents a new method for O(n) planar graph embedding which avoids some of the complexities of previous approaches. The PC-tree method of Shih and Hsu has similarities to our algorithm, but the formulation is incorrect and not O(n) for reasons discussed in this dissertation. Our planarity algorithm operates directly on an adjacency list representation of a collection of planar biconnected components, adding one edge at a time to the embedding until the entire graph is embedded or until a non-planarity condition arises. If the graph is not planar, a new O(n) algorithm is presented that simplifies the extraction of a Kuratowski subgraph (a subgraph homeomorphic to [special characters omitted]). The results are then extended to outerplanar graphs, which are planar graphs that can be embedded with every vertex along the external face. In linear time, the algorithms find an outerplanar embedding or a minimal obstructing subgraph homeomorphic to [special characters omitted]. Finally, modifications to the outerplanarity and planarity obstruction isolators are presented, resulting in O(n) methods for identifying a subgraph homeomorphic to [special characters omitted].