Reducing CMSO to Unbreakable Graphs Cannot be Computable

2026-08-04

AI Summary

This work investigates whether the model-checking problem for Counting Monadic Second-Order (CMSO) logic on graphs admits an efficient reduction to $(q,k)$-unbreakable graphs, with a focus on the computability of the parameter $q$. By integrating techniques from mathematical logic, parameterized complexity, and computability theory, the study demonstrates that $q$ cannot be expressed as a computable function of the CMSO formula. Consequently, any such reduction must necessarily be non-constructive. This result establishes a fundamental barrier to the constructive approach proposed by Lokshtanov et al., revealing an intrinsic limitation in the algorithmic realizability of CMSO model checking.

Abstract

Lokshtanov, Ramanujan, Saurabh, and Zehavi [ICALP 2018] proved that for any CMSO formula $φ$, testing $φ$ on arbitrary graphs can be reduced to testing it on $(q,k)$-unbreakable graphs for appropriate parameters. Their proof is non-constructive, and they ask whether it can be made constructive. We prove that this is impossible: specifically, the parameter $q$ cannot be a computable function of $φ$.

Problem

• CMSO
• unbreakable graphs
• computability
• graph reduction
• parameterized complexity

Innovation

• CMSO
• unbreakable graphs
• computability
• graph reduction
• model checking