# Is the Markov Network (Markov Random Field) property biconditional?

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 $$C$$.