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，有界宽度的团分解可以通过CMSO传递产生。我们证明在锦标赛的情形下，一阶传递就足够了。这意味着在有界线性团宽度的锦标赛上，逻辑CMSO与存在性MSO是等价的。