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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

.