Parameterized Algorithms for Book Embedding Problems

TL;DR

This paper investigates the computational complexity of book embedding problems, showing they are fixed-parameter tractable when parameterized by certain graph measures, thus providing efficient algorithms for specific graph classes.

Contribution

The paper introduces fixed-parameter algorithms for book embedding problems based on vertex cover number and pathwidth, advancing understanding of their computational tractability.

Findings

Fixed-Order Book Thickness is fixed-parameter tractable by vertex cover number.

Book Thickness is fixed-parameter tractable by vertex cover number.

Fixed-Order Book Thickness is fixed-parameter tractable by pathwidth.

Abstract

A k-page book embedding of a graph G draws the vertices of G on a line and the edges on k half-planes (called pages) bounded by this line, such that no two edges on the same page cross. We study the problem of determining whether G admits a k-page book embedding both when the linear order of the vertices is fixed, called Fixed-Order Book Thickness, or not fixed, called Book Thickness. Both problems are known to be NP-complete in general. We show that Fixed-Order Book Thickness and Book Thickness are fixed-parameter tractable parameterized by the vertex cover number of the graph and that Fixed-Order Book Thickness is fixed-parameter tractable parameterized by the pathwidth of the vertex order.