Two-page book embedding of trees under vertex-neighborhood constraints

Abstract

We study the VLSI-related problem of embedding graphs in books. A book embedding of a graph G=( V,E) consists of two parts, namely, (1) an ordering of V along the spine of the book, and (2) an assignment of each eϵ E to a page of the book, so that edges assigned to the same page do not intersect. In devising an embedding, one seeks to minimize the number of pages used. A black/white (b/w) graph is a pair ( G,U), where G is a graph and U⊆ V is a subset of distinguished black vertices (the vertices of V−U are called white). A black/white (b/w) book embedding of a b/w graph ( G,U) is a book embedding of G, where the vertices of U are placed consecutively on the spine. The need for b/w embeddings may arise, for example, when the input ports of a multilayer VLSI chip are to be separated from the output ports. In this paper we prove that every b/w tree admits a two-page b/w embedding. The proof takes the form of a linear time algorithm, which uses an extension of the unfolding technique introduced by Moran and Wolfsthal. Combining this algorithm with the one of Moran and Wolfsthal results in a linear time algorithm for optimal b/w embedding of trees.