Reducing CMSO to Unbreakable Graphs Cannot be Computable

Colin Geniet; Roohani Sharma

Reducing CMSO to Unbreakable Graphs Cannot be Computable

Discrete Mathematics 2026-08-04 v1 Computational Complexity Logic in Computer Science Combinatorics

Authors: Colin Geniet , Roohani Sharma

Abstract

Lokshtanov, Ramanujan, Saurabh, and Zehavi [ICALP 2018] proved that for any CMSO formula $\phi$ , testing $\phi$ 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 $\phi$ .

Cite

@article{arxiv.2608.03144,
  title  = {Reducing CMSO to Unbreakable Graphs Cannot be Computable},
  author = {Colin Geniet and Roohani Sharma},
  journal= {arXiv preprint arXiv:2608.03144},
  year   = {2026}
}

Comments

7 pages to appear at ESA 26