As far as I know, the property of a Markov Random Field is defined as follows:
Let $G = (V, E)$ be a Markov Network. Let $X, Y, C \subseteq V$. If every path from a vertex in $X$ to a vertex in $Y$ passes through a vertex in $C$, then we may say that $X$ and $Y$ are conditionally independent given $C$.
However, I have some conflicting information about whether this statement should be a biconditional (depending on the source I reference). Namely, I'm wondering if the following is true:
If there exists a path from $X$ to $Y$ which does not pass through a vertex in $C$, can we say that $X$ and $Y$ are necessarily not independent given $C$.