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Complexity of the (Connected) Cluster Vertex Deletion problem on $H$-free graphs

Hoang-Oanh Le, Van Bang Le

TL;DR

The paper establishes a complete complexity dichotomy for Cluster Vertex Deletion and its connected variant on $H$-free graphs: these problems are polynomial-time solvable iff $H$ is an induced subgraph of $P_4$, and NP-complete otherwise, with ETH-based subexponential lower bounds in the hard cases. The polynomial cases are proved via $MSOL_1$ expressibility and clique-width arguments on $P_4$-free graphs, yielding linear-time algorithms, while the hardness results come from reductions from Vertex Cover and nae $3$sat to dense and sparse graph classes respectively. The connected variant obeys the same dichotomy, strengthening the set of dichotomy results for well-studied problems on $H$-free graphs. Together, these results significantly advance our understanding of how forbidding induced subgraphs governs the tractability of cluster-based graph modification problems.

Abstract

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.

Complexity of the (Connected) Cluster Vertex Deletion problem on $H$-free graphs

TL;DR

The paper establishes a complete complexity dichotomy for Cluster Vertex Deletion and its connected variant on -free graphs: these problems are polynomial-time solvable iff is an induced subgraph of , and NP-complete otherwise, with ETH-based subexponential lower bounds in the hard cases. The polynomial cases are proved via expressibility and clique-width arguments on -free graphs, yielding linear-time algorithms, while the hardness results come from reductions from Vertex Cover and nae sat to dense and sparse graph classes respectively. The connected variant obeys the same dichotomy, strengthening the set of dichotomy results for well-studied problems on -free graphs. Together, these results significantly advance our understanding of how forbidding induced subgraphs governs the tractability of cluster-based graph modification problems.

Abstract

The well-known Cluster Vertex Deletion problem (CVD) asks for a given graph and an integer whether it is possible to delete a set of at most vertices of such that the resulting graph is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs for which CVD on -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 -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 has to induce a connected subgraph of . It turns out that CCVD admits the same complexity dichotomy for -free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on -free graphs.
Paper Structure (9 sections, 24 theorems, 15 equations, 4 figures, 1 algorithm)

This paper contains 9 sections, 24 theorems, 15 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction and results
  2. Preliminaries
  3. $H$-free graphs: polynomial cases
  4. Cluster-VD and Connected Cluster-VD on dense graphs
  5. Cluster-VD and Connected Cluster-VD on sparse graphs
  6. $H$-free graphs: $\mathsf{NP}$-completeness cases
  7. Conclusion
  8. Computing the cluster vertex deletion number of cographs using the cotrees
  9. Computing the connected cluster vertex deletion number of cographs using the cotrees

Key Result

Theorem 1

Let $H$ be a fixed graph. cluster-vd is polynomially solvable on $H$-free graphs if $H$ is an induced subgraph of the $4$-vertex path $P_4$, and $\mathsf{NP}$-complete otherwise. Furthermore, in case $H$ is not an induced subgraph of $P_4$, no algorithm of runtime $2^{o(n)}$ can solve cluster-vd on

Figures (4)

  • Figure 1: The graphs $H$ for which cluster-vd and connected cluster-vd are polynomially solvable on $H$-free graphs.
  • Figure 3: The tree $H(g,r,s)$. The $(g+2)n$ black vertices form an optimal (connected) cluster vertex deletion set.
  • Figure 4: An example of the reduction from cluster-vd to connected cluster-vd: A bipartite graph $G$ (left) and the bipartite graph $G(3)$ (right) obtained from $G$ and $H(3,4,3)$; the bipartition of the vertex set is indicated by circle and rectangle vertices.
  • Figure 5: A cograph $G$, the cotree of $G$ and its binary version.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Moret98ImpagliazzoPZ01
  • Theorem 4: Tovey84Moret98ImpagliazzoPZ01
  • Proposition 5
  • Theorem 6: JohnsonS99Murphy92Komusiewicz18
  • Theorem 7
  • Theorem 8
  • Theorem 9: Yannakakis81a
  • Theorem 10: HsiehLLP22
  • ...and 15 more