Transducing Linear Decompositions of Tournaments

TL;DR

This paper demonstrates that for tournaments with bounded linear clique-width, first-order transductions can produce clique-decompositions, establishing the equivalence of CMSO and existential MSO logics in this context.

Contribution

It shows that first-order transductions suffice for tournaments with bounded linear clique-width, simplifying the logical framework needed.

Findings

First-order transductions are sufficient for bounded linear clique-width tournaments.

CMSO and existential MSO are equivalent over these tournaments.

Simplifies logical descriptions of such tournaments.

Abstract

Boja\'nczyk, Pilipczuk, and Grohe [LICS '18] proved that for graphs of bounded linear clique-width, clique-decompositions of bounded width can be produced by a CMSO transduction. We show that in the case of tournaments, a first-order transduction suffices. This implies that the logics CMSO and existential MSO are equivalent over bounded linear clique-width tournaments.