Optimum embedding of complete graphs in books Original Research Article

Embedding a graph in a book is an arrangement of vertices in a line along the spine of the book and edges on the pages in such a way that edges residing on the same page do not cross. Each nontrivial graph has many different embeddings in books. The embedding with minimum number of pages and with minimum width is optimal. Chung et al. in Embedding graphs in books: a layout problem with applications to VLSI design (1987), described a general method for embedding the complete graph Kn in a book of width n with ⌈n/2⌉ pages. The paper presents a simple and effective method for optimum embedding of width n − 3 of any complete graph Kn in a book with n/2 pages for even n and (n + 1)/2 pages for odd n. It is also proved that the set of complete graphs is a proper subset of the set of graphs for which the presented algorithm gives an optimal number of pages.