Book Embeddings of Graph Products

TL;DR

This paper investigates the stack layout properties of graph products, establishing bounds on the stack number for certain graph classes using a novel simultaneous stack-queue layout concept.

Contribution

It introduces new bounds on the stack number for strong products of paths with graphs of bounded pathwidth or bounded degree, via a novel simultaneous layout approach.

Findings

Stack number is bounded for strong product of a path and a graph with bounded pathwidth.

Stack number is bounded for strong product of a path and a bipartite graph with bounded treewidth and degree.

Introduces the concept of simultaneous stack-queue layouts, which may have broader applications.

Abstract

A $k$-stack layout (also called a $k$-page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into $k$ sets of non-crossing edges with respect to the vertex order. The stack number (book thickness, page number) of a graph is the minimum $k$ such that it admits a $k$-stack layout. A $k$-queue layout is defined similarly, except that no two edges in a single set may be nested. It was recently proved that graphs of various non-minor-closed classes are subgraphs of the strong product of a path and a graph with bounded treewidth. Motivated by this decomposition result, we explore stack layouts of graph products. We show that the stack number is bounded for the strong product of a path and (i) a graph of bounded pathwidth or (ii) a bipartite graph of bounded treewidth and bounded degree. The results are obtained via a novel concept of simultaneous stack-queue layouts, which may be of independent interest.