Simpler algorithms for testing 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 with a fixed partition of the edge set (called a partitioned graph) admits a 2-page book embedding such that each edge subset in the partition is embedded in one of the two pages. We first show that finding a 2-page book embedding of a partitioned graph can be reduced to the planarity testing of a graph, which yields a simple linear-time algorithm for solving the problem. We then characterize the instances of partitioned graphs that do not admit 2-page book embeddings via forbidden subgraphs, and give a linear-time algorithm for detecting a forbidden subgraph of a given partitioned graph. As an application of our main result, we also show that our book embedding results imply an alternative proof that the clustered planarity problem with two clusters can be solved in linear time. (C) 2016 Elsevier B.V. All rights reserved.