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The condition $C(p)$ (resp. $C'(p)$) is: For any set $S$ of $p$ vertices in $D$, there exists $x \\in S$ such that $N^+_D(x) \\subseteq N^+_D(y)$ (resp. $N^-_D(x) \\subseteq N^-_D(y)$) for all $y \\in S$, where $N^+_D(x)$ (resp. $N^-_D(x)$) is the set of out-neighbors (resp. in-neighbors) of $x$ in $D$. 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