Bandwidth Parameterized by Cluster Vertex Deletion Number

TL;DR

This paper investigates the parameterized complexity of the Bandwidth problem, showing it is fixed-parameter tractable when combining cluster vertex deletion number and clique number, but W[1]-hard when parameterized solely by cluster vertex deletion number.

Contribution

It establishes new fixed-parameter tractability and hardness results for Bandwidth based on cluster vertex deletion number, advancing understanding of its parameterized complexity.

Findings

FPT when parameterized by cluster vertex deletion number plus clique number

W[1]-hard when parameterized only by cluster vertex deletion number

Narrowed complexity gaps in Bandwidth problem analysis

Abstract

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 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 $\omega$, 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.