Embedding 5-planar graphs in three pages

TL;DR

This paper presents an efficient algorithm for embedding 5-planar graphs into a three-page book, improving the understanding of graph embedding complexity and providing practical methods for applications in VLSI and parallel processing.

Contribution

It introduces an $O(n^{2})$ time algorithm for embedding 5-planar graphs into three pages, advancing previous results on planar graph embeddings.

Findings

Successfully embedded 5-planar graphs in three pages

Achieved $O(n^{2})$ time complexity for the embedding algorithm

Extended the understanding of graph embedding page numbers

Abstract

A \emph{book-embedding} of a graph $G$ is an embedding of vertices of $G$ along the spine of a book, and edges of $G$ on the pages so that no two edges on the same page intersect. the minimum number of pages in which a graph can be embedded is called the \emph{page number}. The book-embedding of graphs may be important in several technical applications, e.g., sorting with parallel stacks, fault-tolerant processor arrays design, and layout problems with application to very large scale integration (VLSI). Bernhart and Kainen firstly considered the book-embedding of the planar graph and conjectured that its page number can be made arbitrarily large [JCT, 1979, 320-331]. Heath [FOCS84] found that planar graphs admit a seven-page book embedding. Later, Yannakakis proved that four pages are necessary and sufficient for planar graphs in [STOC86]. Recently, Bekos et al. [STACS14] described an $O(n^{2})$ time algorithm of two-page book embedding for 4-planar graphs. In this paper, we embed 5-planar graphs into a book of three pages by an $O(n^{2})$ time algorithm.