Page Number and Graph Treewidth

Page Number and Graph Treewidth

Li Xianglu (Zhongyuan University of Technology, China)
DOI: 10.4018/978-1-4666-0270-0.ch018
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Book-embedding of graph G involves embedding its vertices along the spine of the book and assigning its edges to pages of the book such that no two edges cross on the same page. The pagenumber of G is the minimum number of pages in a book-embedding of G. In this paper, the authors also examine the treewidth TW(G), which is the minimum k such that G is a subgraph of a k-tree. The authors then study the relationship between pagenumber and treewidth. Results show that PN(G)=TW(G), which proves a conjecture of Ganley and Heath showing that some known upper bounds for the pagenumber can be improved.
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The book-embedding problem for graph G 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 G=(V,E) with n vertices, a bijection f:978-1-4666-0270-0.ch018.m01is called a labeling of G by Chung(1988), where 978-1-4666-0270-0.ch018.m02 represents the label of vertex v ϵ V. Let 978-1-4666-0270-0.ch018.m03be the vertex with label i. Then the labeling f can also be regarded as an ordering 978-1-4666-0270-0.ch018.m04 on a line. For a labeling f, two edges 978-1-4666-0270-0.ch018.m05 are said to be crossing if 978-1-4666-0270-0.ch018.m06 or 978-1-4666-0270-0.ch018.m07.

With respect to a labeling f, a partition 978-1-4666-0270-0.ch018.m08 of the edge set E(G) is called a page partition if no two edges in any subset 978-1-4666-0270-0.ch018.m09 are crossing. This page partition can be thought of as a coloring of E(G) where the edges in Ei have color i and no two edges of the same color are crossing. Thus, a page partition C represents an assignment of edges of G to pages of the book. We call the minimum number of subsets in a page partition C the page number of G under labeling f, and denote it by PN(G,f). The pagenumber of G is then defined as

where f is taken over all labelings of G.

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