Bojańczyk, 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.

翻译：Bojańczyk、Pilipczuk 和 Grohe [LICS '18] 证明了，对于有界线性团宽度的图，可以通过 CMSO 转导生成有界宽度的团分解。我们证明，在锦标赛的情况下，一阶转导就足够了。这意味着在具有有界线性团宽度的锦标赛上，CMSO 逻辑和存在性 MSO 逻辑是等价的。