On parameterized algorithms for fixed-order book thickness with respect to the pathwidth of the vertex ordering

Abstract

Given a graph G=(V,E) with a vertex ordering ≺, the fixed-order book thickness problem asks whether there is a page assignment σ such that 〈≺,σ〉 is a k-page book embedding of G. This problem is NP-complete even for any fixed k greater than 3. Recently, Bhore et al. (2019, 2020) [1,2] presented a parameterized algorithm with respect to the pathwidth κ of the vertex ordering. In this paper, we first re-analyze the running time for Bhore et al.'s algorithm, and prove a bound of 2O(κ2)⋅|V| improving Bhore et al.'s bound of κO(κ2)⋅|V|. Then, we show that fixed-order book thickness parameterized by the pathwidth of the vertex ordering does not admit a polynomial kernel unless NP ⊆ coNP/poly. Finally, we show that a generalized fixed-order book thickness problem, in which a budget of at most c crossings over all pages was given, admits a parameterized algorithm running in time (c+2)O(κ2)⋅|V|.