Reducing CMSO to Unbreakable Graphs Cannot be Computable

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 $\phi$.