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Introduction
The book-embedding problem for graph 
is to embed its vertices onto a line along the spine of the book and to draw the edges on pages of the book such that no two edges on the same page cross, and the number of used pages is minimized.
The book-embedding problem has been motivated by several areas of computer science such as VLSL theory, multilayer printed circuit boards (PCB), sorting with parallel stacks and Turning-machine and the design of fault-tolerant processor arrays, etc (e.g., Chung et al., 1987). The DIOGENES approach to fault-tolerant processor arrays, proposed by Rosenberg (1986), is the most famous one. In the DIOGENES approach, the processing elements are laid out in a logical line, and some number of bundles of wires run in parallel with the line. The faulty elements are bypassed, and the fault-free ones are interconnected through the bundles. Here, the bundles work as queues and/or stacks. If the bundles work as stacks, then the realization of an interconnection network needs a book-embedding of the interconnection network. In this case, the number of pages corresponds to the number of bundles of the DIOGENES stack layout. Therefore, book-embeddings with few pages realize more hardware-efficient DIOGENES stacks layouts.
The book-embedding problem can be stated as a graph-labeling problem as follows. We shall follow the graph-theoretic terminology and notation used by Bondy and Murty (1976) and Golumbic (1980).
Given a simple connected graph 
with 
vertices, a bijection f:
is called a labeling of 
by Chung(1988), where 
represents the label of vertex 
Let 
be the vertex with label 
Then the labeling f can also be regarded as an ordering 
on a line. For a labeling f, two edges 
are said to be crossing if 
or 
.
With respect to a labeling
, a partition 
of the edge set 
is called a page partition if no two edges in any subset 
are crossing. This page partition can be thought of as a coloring of 
where the edges in 
have color 
and no two edges of the same color are crossing. Thus, a page partition 
represents an assignment of edges of 
to pages of the book. We call the minimum number of subsets in a page partition 
the page number of 
under labeling 
, and denote it by
. The pagenumber of 
is then defined as
where
is taken over all labelings of
.