Simpler Testing for Two-page Book Embedding of Partitioned Graphs

A 2-page book embedding of a graph is to place the vertices linearly on a spine (a line segment) and the edges on the two pages (two half planes sharing the spine) so that each edge is embedded in one of the pages without edge crossings. Testing whether a given graph admits a 2-page book embedding is known to be NP-complete. In this paper, we study the problem of testing whether a given graph admits a 2-page book embedding with a fixed edge partition. We first show that finding a 2-page book embedding of a given graph can be reduced to the planarity testing of a graph, which yields a simple linear-time algorithm for solving the problem. We also characterize the graphs that do not admit 2-page book embeddings via forbidden subgraphs, and give a linear-time algorithm for detecting the forbidden subgraph of a given graph.