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

2026-08-04

AI Summary

This study investigates the parameterized complexity of Induced Subgraph Isomorphism (ISI) and Maximum Common Induced Subgraph (MCIS) with respect to the cluster vertex deletion number. By employing algebraic reductions to Exact Multicolored Matching and its weighted variant, combined with randomized algorithms and lower-bound analysis under the Exponential Time Hypothesis (ETH), the authors establish for the first time that ISI is fixed-parameter tractable under this parameter, yielding an $O^*(k^{O(k)})$-time algorithm that is ETH-optimal. In contrast, they demonstrate that MCIS is strictly harder than ISI: they present an $O^*(2^{O(k^2)})$-time algorithm for MCIS and prove its tightness under ETH, while also showing that 3-MCIS remains NP-hard even when the cluster vertex deletion number is merely 2, thereby revealing a fundamental complexity gap between the two problems.

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.

Problem

• Induced Subgraph Isomorphism
• Maximum Common Induced Subgraph
• parameterized complexity
• cluster vertex deletion number
• 3-MCIS

Innovation

• induced subgraph isomorphism
• maximum common induced subgraph
• cluster vertex deletion number
• fixed-parameter tractability
• Exponential Time Hypothesis