Cluster Vertex Deletion Problems on Cubic Graphs

TL;DR

This paper proves that Cluster Vertex Deletion and s-Club Vertex Deletion problems are NP-complete for cubic planar bipartite graphs, extending the understanding of their computational complexity in specific graph classes.

Contribution

It establishes NP-completeness of Cluster-VD and s-Club-VD for cubic planar bipartite graphs, and provides new NP-completeness results for k-Path Vertex Cover in related graph classes.

Findings

NP-completeness of Cluster-VD for cubic planar bipartite graphs

NP-completeness of 3-PVC for cubic planar bipartite graphs

NP-completeness of k-PVC for large girth subcubic planar 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.