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 $〈\prec ,\sigma 〉$ 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.(GD 2019; J. Graph Algorithms Appl. 2020) 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 $2^{\mathcal{O}({\kappa }^2)}\cdot |V|$ improving Bhore et al.'s bound of ${\kappa }^{\mathcal{O}({\kappa }^2)}\cdot |V|$. 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)}^{\mathcal{O}({\kappa }^2)}$$\cdot |V|$.

给定一个图 $G=（\mathrm{伏特}，E）$在顶点顺序为≺的情况下，固定顺序的书本厚度问题询问是否存在页面分配σ，使得$〈\prec ，\sigma 〉$是嵌入G的k页图书。即使对于任何大于3的固定k，这个问题也是NP完全的。最近，Bhore等人（GD 2019； J。Graph Algorithms Appl。2020）提出了一种关于顶点排序的路径宽度κ的参数化算法。在本文中，我们首先重新分析了Bhore等人算法的运行时间，并证明了${\mathrm{2个}}^{\mathcal{Ø}（{\kappa }^{\mathrm{2个}}）}\cdot |\mathrm{伏特}|$ 改善Bhore等人的界限 ${\kappa }^{\mathcal{Ø}（{\kappa }^{\mathrm{2个}}）}\cdot |\mathrm{伏特}|$。然后，我们表明，除非NP⊆coNP / poly，否则由顶点排序的路径宽度参数化的固定顺序的书本厚度不容许多项式核。最后，我们证明了一个广义的固定顺序书本厚度问题，其中给出了所有页面上最多c个交叉点的预算，它接受了及时运行的参数化算法${（C+\mathrm{2个}）}^{\mathcal{Ø}（{\kappa }^{\mathrm{2个}}）}$$\cdot |\mathrm{伏特}|$。