One-page book embedding under vertex-neighborhood constraints

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 eE 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 minimi;ze the number of pages used. This paper addresses a generalization of the book embedding problem, called the black/white (blw) book embedding problem, where a vertex-neighborhood constraint is imposed on the ordering of the ver­ tices. A blw graph is a pair (G ,V), where G=(V ,E) is a graph and V c;V is a set of distinguished black venices (the vertices in V-V are called white). A blw book embedding of (G ,V) is a book embedding of G, where the black vertices are arranged consecutively along the spine. Given a b/w graph (G ,V), the blw book embedding problem is that of finding a b/w book embedding of (G ,V), such that number of pages is minimized. The need for b/w embeddings may arise, for example, in applications where the input ports of a VLSI chip arc to be separated from the output ports. The main res91t of this paper is a characterization of the b/w graphs that admit a one-page b/w embedding. The characterization is given in tenns of a set of forbidden blw subgraphs, the absence of which is necessary and sufficient for one-page b/w embedding. For a b/w graph with non~ of these forbid­ den subgraphs, a one-page b/w embedding is constructible in linear tUne. The construction utilizes a tech­ nique called blw unfolding, which is a feature of independent interest