Cluster Vertex Deletion Problems on Cubic Graphs
Abstract
The problems Cluster Vertex Deletion (or Cluster-VD) and its generalization s-Club Cluster Vertex Deletion (or s-Club-VD, for any integer s>= 1), have been introduced with the aim of detecting highly-connected parts in complex systems. Their NP-completeness has been established for several classes of graphs, but remains open for smaller classes, including subcubic planar bipartite graphs and cubic graphs. In this paper, we show that Cluster-VD and more generally s-Club-VD are NP-complete for cubic planar bipartite graphs. We also deduce new results for the related k-Path Vertex Cover problem (or k-PVC), namely 3-PVC is NP-complete for cubic planar bipartite graphs, whereas k-PVC with k>= 4 is NP-complete for subcubic planar (and bipartite, when k is odd) graphs of arbitrarily large girth.
Cite
@article{arxiv.2505.07443,
title = {Cluster Vertex Deletion Problems on Cubic Graphs},
author = {Irena Rusu},
journal= {arXiv preprint arXiv:2505.07443},
year = {2025}
}
Comments
21 pages, 6 figures