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 $b$，Bandwidth 问题询问是否存在从 $V(G)$ 到 $\{1, \ldots, |V(G)|\}$ 的双射 $\pi$，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 的树和发长为 3 的毛毛虫图）中仍保持 NP 完全性。在参数化复杂性领域，这些结果意味着该问题在有界路径宽度的图上仍然是 NP 难的，此外已知当以输入图的树深度为参数时它是 W[1]-难的。相反，当以输入图的顶点覆盖数为参数时，该问题确实变为 FPT 的。在本文中，我们推进了 Bandwidth 参数化（不）可解性的研究。我们首先表明，当以团簇顶点删除数 cvd 加上输入图的团数 $\omega$ 为参数时，它是 FPT 的，从而推广了前述关于顶点覆盖的结果。另一方面，我们证明当仅以 cvd 为参数时，Bandwidth 是 W[1]-难的。我们的结果推广了一些先前的结果，并缩小了部分复杂性差距。