In this note, we introduce the \emph{partial order decomposition number} of a digraph $D$, denoted $pod(D)$, defined as the minimum integer $k$ such that $A(D)=A(P_1)\cup\cdots\cup A(P_k)$, where $P_1,\ldots,P_k$ are partial orders on $V(D)$. We prove that $\dic(D)\le \diomega(D)^{pod(D)}$ for every digraph $D$. In particular, every class of digraphs with bounded $pod$ is polynomially $\dic$-bounded. We apply this to tournaments, showing that if $\mathcal C$ is a class of tournaments with bounded dichromatic number, then the closure of $\mathcal C$ under substitution is polynomially $\dic$-bounded, thereby making progress on a question of Aubian, Charbit, Lopes, and the first author. As further applications of $pod$, we prove that poset tournaments of bounded dimension are $\dic$-bounded, derive polynomial lower bounds on the directed clique number of an explicit family of tournaments, thereby answering a conjecture of Gutowski and Rams, and show that tournaments with bounded $pod$ have bounded domination number.

翻译：在本文中，我们引入了有向图$D$的\emph{偏序分解数}$pod(D)$，定义为满足$A(D)=A(P_1)\cup\cdots\cup A(P_k)$的最小整数$k$，其中$P_1,\ldots,P_k$是$V(D)$上的偏序。我们证明了对每个有向图$D$，有$\dic(D)\le \diomega(D)^{pod(D)}$成立。特别地，每个具有有界$pod$的有向图类都是多项式$\dic$有界的。我们将此应用于锦标赛，证明若$\mathcal C$是具有有界二色数的锦标赛类，则$\mathcal C$在替换下的闭包是多项式$\dic$有界的，从而推进了Aubian、Charbit、Lopes和第一作者提出的一个问题。作为$pod$的进一步应用，我们证明了有界维数的偏序集锦标赛是$\dic$有界的，推导了一个显式锦标赛族的有向团数的多项式下界，从而回答了Gutowski和Rams的一个猜想，并证明具有有界$pod$的锦标赛具有有界支配数。