Complexity of induced subgraph isomorphism and maximum common induced subgraph parameterized by cluster vertex deletion number

Tomohiro Koana; Soh Kumabe; Yota Otachi

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arXiv · 2608.03845

Complexity of induced subgraph isomorphism and maximum common induced subgraph parameterized by cluster vertex deletion number

Tomohiro Koana·Soh Kumabe·Yota Otachi

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Abstract

We study the parameterized complexity of Induced Subgraph Isomorphism (ISI) and Maximum Common Induced Subgraph (MCIS) with respect to the cluster vertex deletion number $k$. For ISI, we give a randomized $O^*(k^{O(k)})$-time algorithm, showing that ISI is fixed-parameter tractable under this parameter and resolving an open question of Hanaka et al. [WALCOM 2026]. Our algorithm is optimal under the Exponential Time Hypothesis (ETH), and is based on a reduction to Exact Multicolored Matching solvable via algebraic techniques. For MCIS, we present a randomized $O^*(2^{O(k^2)})$-time algorithm via a reduction to a weighted variant of Exact Multicolored Matching, and we prove a matching ETH-based lower bound by showing that a $k$-by-$k$ binary matrix feasibility problem with list-constrained rows and columns admits no $O^*(2^{o(k^2)})$-time algorithm, which may be of independent interest. These results reveal that, in this setting, MCIS is strictly harder than ISI. Finally, for the three-graph variant 3-MCIS, we show that it becomes NP-hard already when each input graph has cluster vertex deletion number 2.

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Tomohiro Koana, Soh Kumabe, Yota Otachi. 2026-08-04. Complexity of induced subgraph isomorphism and maximum common induced subgraph parameterized by cluster vertex deletion number. https://arxiv.org/abs/2608.03845

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