In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not\to H$. The directed path on $k+1$ vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair. This relation between paths and tournaments implies that a graph is $k$-colourable if and only if it admits an orientation with no directed path on more than $k$-vertices. In this work, for every undirected cycle $C$ we find an orientation $C_D$ and an oriented path $P_C$, such that $(P_C,C_D)$ is a duality pair. As a consequence we obtain that there is a finite set, $F_C$, such that an undirected graph is homomorphic to $C$, if and only if it admits an $F_C$-free orientation. As a byproduct of the proposed duality pairs, we show that if $T$ is a tree of height at most $3$, one can choose a dual of $T$ of linear size with respect to the size of $T$.

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有向环和无向环的对偶对和同态

在有向图的同态序中，对偶对是有序的有向图对 $(G,H)$ 使得对于任何有向图，$D$, $G\to D$ 当且仅当 $D\not\to H $. $k+1$ 顶点上的有向路径以及 $k$ 顶点上的传递锦标赛是对偶对的经典示例。路径和锦标赛之间的这种关系意味着一个图是 $k$-可着色的，当且仅当它承认在超过 $k$-个顶点上没有定向路径的方向。在这项工作中，对于每个无向环 $C$，我们找到了一个方向 $C_D$ 和一个有向路径 $P_C$，这样 $(P_C,C_D)$ 是一个对偶对。因此，我们得到一个有限集 $F_C$，使得一个无向图同态于 $C$，当且仅当它承认无 $F_C$ 的方向。作为提议的对偶对的副产品，