One-page book embedding under vertex-neighborhood constraints

The authors address a generalization of the book embedding problem, called the black/white (b/w) book embedding problem, where a vertex-neighborhood constraint is imposed on the ordering of the vertices. A b/w graph is a pair (G,U), where G=(V,E) is a graph and U⊆V is a set of distinguished black vertices (the vertices in V -U are called white). A b/w book embedding of (G, U) is a book embedding of G, where the black vertices are arranged consecutively along the spine. Given a b/w graph (G,U) the b/w book embedding problem is that of finding a b/w book embedding of (G,U) such that number of pages is minimized. The main result is a characterization of the b/w graphs that admit a one-page b/w embedding. The characterization is given in terms of a set of forbidden b/w subgraphs, the absence of which is necessary and sufficient for one-page b/w embedding. For a b/w graph with none of these forbidden subgraphs, a one-page b/w embedding can be constructed in linear time. The construction utilizes a technique called b/w unfolding, which is a feature of independent interest