Parameterized Algorithms for Book Embedding Problems

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.

Grid Recognition: Classical and Parameterized Computational Perspectives

“…Most (in particular, the early efforts) have been directed at variants of the classic Crossing Minimization problem, introduced by Turán in 1940 [56], parameterized by the number of crossings (see, e.g., [34,46,26,39,40,47]). However, in the past few years, there is an increasing interest in the analysis of a variety of other problems in graph drawing from the perspective of parameterized complexity (see, e.g., [10,2,35,14,38,6,21,20,11,23,49,48] and the upcoming Dagstuhl seminar [1]).…”