Cluster Vertex Deletion Problems on Cubic Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913832552628224 |
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| author | Rusu, Irena |
| author_facet | Rusu, Irena |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07443 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cluster Vertex Deletion Problems on Cubic Graphs Rusu, Irena Computational Complexity 68Q25, 68R10 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. |
| title | Cluster Vertex Deletion Problems on Cubic Graphs |
| topic | Computational Complexity 68Q25, 68R10 |
| url | https://arxiv.org/abs/2505.07443 |