On Fixed-Order Book Thickness Parameterized by the Pathwidth of the Vertex Ordering

Abstract

Given a graph $G=(V,E)$ and a fixed linear order $\prec$ of V, the problem fixed-order book thickness asks whether there is a page assignment $\sigma$ such that $\langle \prec ,\sigma \rangle$ is a k-page book embedding of G. Recently, Bhore et al.(GD2019) presented an algorithm parameterized by the pathwidth of the vertex ordering (denoted by $\kappa$). In this paper, we first re-analyze the running time for Bhore et al.’s algorithm, and prove a bound of $2^{O(\kappa ^2)}\cdotp |V|$ improving on Bhore et al.’s bound of $\kappa ^{O(\kappa ^2)}\cdotp |V|$. We further show that this parameterized problem does not admit a polynomial kernel unless NP $\subseteq$ coNP/poly. Finally, we show that the general fixed-order book thickness problem, in which a budget of at most c crossings over all pages was given, admits an algorithm running in time $(c+2)^{O(\kappa ^2)}$ $\cdotp |V|$.

This research was supported in part by the National Natural Science Foundation of China under Grant (No. 61572190, 61972423), 111 Project (No. B18059) and Hunan Provincial Science and Technology Program (No. 2018TP1018, 2018WK4001).