Algorithms for Embedding Graphs in Books

We investigate the problem ol embedding graphs in boob. A book is some number or halfplanes (the page• or the book), which share a common line as boundary (the qine or the book). A book embedding or a graph embeds the vertices on the spine in some order and embeds each edge in some page so that in each page no two edges intersect. The pagenumber ol a graph is the number or pages in a minimum-page embedding or the graph. The pagewidth or a book embedding is the maximum cutwidth or the embedding in any page. A practical application is in the realization or a fault-tolerant array or VLSI processors. Our results are efficient algorithms for embedding certain classes or planar graphs in books or small pagenumber or small pagewidth. The first result is a linear time algorithm that embeds any planar graph in a book or seven pages. This establishes the smallest upper bound known for the pagenumber or the class or planar graphs. The algorithm uses three main ideas. The first is to level the planar graph. The second is to eztend a cycle at one level to the next level by doing micro-surgery. The third is to neat the embedding or successive levels to obtain finite pagenumber. The second result is a linear time algorithm that embeds any trivalent planar graph in a book or two pages. The algorithm edge-augments the graph to make it hamiltonian while keeping it planar. The third result is an 0( n logn) time algorithm for embedding any outerplanar graph with small pagewidth. Our algorithm embeds any ,._valent outerplanar graph in a two-page boolr. with O(dlogn) pagewidth. This result is optimal in pagewidth to within a constant factor. The significance for VLSI design is that any outerplanar graph can be implemented in small area in a fault-tolerant fashion.