We present a new, efficient procedure to establish Markov equivalence between directed graphs that may or may not contain cycles under the \textit{d}-separation criterion. It is based on the Cyclic Equivalence Theorem (CET) in the seminal works on cyclic models by Thomas Richardson in the mid '90s, but now rephrased from an ancestral perspective. The resulting characterization leads to a procedure for establishing Markov equivalence between graphs that no longer requires tests for d-separation, leading to a significantly reduced algorithmic complexity. The conceptually simplified characterization may help to reinvigorate theoretical research towards sound and complete cyclic discovery in the presence of latent confounders. This version includes a correction to rule (iv) in Theorem 1, and the subsequent adjustment in part 2 of Algorithm 2.

翻译：我们提出了一种高效的新方法，用于在可能包含环的有向图中，基于d-分离准则建立马尔可夫等价性。该方法源于Thomas Richardson在20世纪90年代中期关于环状模型的经典工作中提出的环等价定理（CET），但本文从祖先视角对其重新表述。由此得到的刻画方法不再需要测试d-分离条件，从而显著降低了算法复杂度。这一概念上的简化刻画可能有助于重新激发关于存在隐变量时环状因果发现的完整性与正确性理论研究的活力。本版本修正了定理1中规则（iv）的表述，并对算法2的第二部分进行了相应调整。