Given a graph $G$ and an integer $b$, Bandwidth asks whether there exists a bijection $\pi$ from $V(G)$ to $\{1, \ldots, |V(G)|\}$ such that $\max_{\{u, v \} \in E(G)} | \pi(u) - \pi(v) | \leq b$. This is a classical NP-complete problem, known to remain NP-complete even on very restricted classes of graphs, such as trees of maximum degree 3 and caterpillars of hair length 3. In the realm of parameterized complexity, these results imply that the problem remains NP-hard on graphs of bounded pathwidth, while it is additionally known to be W[1]-hard when parameterized by the treedepth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number of the input graph. In this paper, we make progress towards the parameterized (in)tractability of Bandwidth. We first show that it is FPT when parameterized by the cluster vertex deletion number cvd plus the clique number $\omega$ of the input graph, thus generalizing the previously mentioned result for vertex cover. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results generalize some of the previous results and narrow some of the complexity gaps.

翻译：给定图$G$和整数$b$，带宽问题询问是否存在一个双射$\pi$从$V(G)$到$\{1, \ldots, |V(G)|\}$，使得$\max_{\{u, v \} \in E(G)} | \pi(u) - \pi(v) | \leq b$。这是一个经典的NP完全问题，已知即使在非常受限的图类（如最大度为3的树和毛长为3的毛毛虫图）上仍保持NP完全性。在参数化复杂性领域，这些结果表明该问题在路径宽度有界的图上仍是NP困难的，同时已知当以输入图的树深度为参数时问题是W[1]-困难的。相比之下，当以输入图的顶点覆盖数为参数时，该问题变为FPT。在本文中，我们在带宽问题的参数化（难）解性方面取得了进展。我们首先证明，当以簇顶点删除数cvd加上输入图的团数$\omega$为参数时，问题是FPT的，从而推广了之前关于顶点覆盖的结果。另一方面，我们证明仅以cvd为参数时，带宽问题是W[1]-困难的。我们的结果推广了部分先前的结果，并缩小了一些复杂性差距。