Complexity of the (Connected) Cluster Vertex Deletion problem on $H$-free graphs
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866914670384775168 |
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| author | Le, Hoang-Oanh Le, Van Bang |
| author_facet | Le, Hoang-Oanh Le, Van Bang |
| contents | The well-known Cluster Vertex Deletion problem (CVD) asks for a given graph $G$ and an integer $k$ whether it is possible to delete a set $S$ of at most $k$ vertices of $G$ such that the resulting graph $G-S$ is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs $H$ for which CVD on $H$-free graphs is polynomially solvable and for which it is NP-complete. Moreover, in the NP-completeness cases, CVD cannot be solved in sub-exponential time in the vertex number of the $H$-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of CVD, the Connected Cluster Vertex Deletion problem (CCVD), in which the set $S$ has to induce a connected subgraph of $G$. It turns out that CCVD admits the same complexity dichotomy for $H$-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on $H$-free graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04931 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complexity of the (Connected) Cluster Vertex Deletion problem on $H$-free graphs Le, Hoang-Oanh Le, Van Bang Discrete Mathematics Computational Complexity Data Structures and Algorithms Combinatorics The well-known Cluster Vertex Deletion problem (CVD) asks for a given graph $G$ and an integer $k$ whether it is possible to delete a set $S$ of at most $k$ vertices of $G$ such that the resulting graph $G-S$ is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs $H$ for which CVD on $H$-free graphs is polynomially solvable and for which it is NP-complete. Moreover, in the NP-completeness cases, CVD cannot be solved in sub-exponential time in the vertex number of the $H$-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of CVD, the Connected Cluster Vertex Deletion problem (CCVD), in which the set $S$ has to induce a connected subgraph of $G$. It turns out that CCVD admits the same complexity dichotomy for $H$-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on $H$-free graphs. |
| title | Complexity of the (Connected) Cluster Vertex Deletion problem on $H$-free graphs |
| topic | Discrete Mathematics Computational Complexity Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2402.04931 |