An Embedding Algorithm for a Specialcase of Extended Grids

Abstract

Abstract:A book consists of a line in the 3-dimensional space, called the spine, and a number of pages, each a half-plane with the spine as boundary. A book embedding (, ) of a graph consists of a linear ordering of , of vertices, called the spine ordering, along the spine of a book and an assignment , of edges to pages so that edges assigned to the same page can be drawn on that page without crossing. That is, we cannot find vertices u, v, x, y with (u) < (x) < (v) < (y), yet the edges uv and xy are assigned to the same page, that is (uv) = (xy). The book thickness or page number of a graph G is the minimum number of pages in required to embed G in a book. In this paper we consider the extended grid and prove that the 1n extended grid can be embedded in two pages. We also give a linear time algorithm to embed the 1n extended grid in two pages.