Bandwidth Parameterized by Cluster Vertex Deletion Number

Given a graph G and an integer b, Bandwidth asks whether there exists a bijection pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} from V(G) to {1,& mldr;,|V(G)|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1, \ldots , |V(G)|\}$$\end{document} such that max{u,v}is an element of E(G)|pi(u)-pi(v)|<= b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max _{\{u, v \} \in E(G)} | \pi (u) - \pi (v) | \le b$$\end{document}. 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 tree-depth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number. In this paper we make progress in understanding 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}, thus significantly strengthening the previously mentioned result for vertex cover number. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results develop and generalize some of the methods of argumentation of the previous results and narrow some of the complexity gaps.