Conditional independence, partial correlation

Question

In my work, I am modelling graphs by measuring the zero- and first-order conditional independence between the variables. That is, if there are three variables, say $A$,$B$ and $C$, an edge between the variables $A$ and $B$ is drawn in the graph if and only if, zero- and first-order correlations between these two variables both differ from zero, that is, if

1. Partial correlation between $A$ and $B$ given $C$ is greater than 0 , that is, $A$ and $B$ are correlated and the correlation between A and B can not be explained by C.

2. correlation between $A$ and $B$ is greater than 0.

In the code I took from someone else, there is only the test for partial correlation (first-order conditional independence) but there is no test for correlation (zero order correlation). Is it safe to assume, that if, If there is correlation between $A$ and $B$ given $C$, it automatically implies that there is correlation between $A$ and $B$? For zero and first order partial correlations, is it enough to test for first order only? Or both are needed ?

Answer 1

Is it safe to assume, that if, If there is correlation between 𝐴 and 𝐵 given 𝐶, it automatically implies that there is correlation between 𝐴 and 𝐵?

No, let $A$ be $1$ if a fair coin's first toss is heads and $0$ o/w and $B$ be 1 if the coin's second toss is heads and $0$ o/w. Let $C$ be the number of heads in the two tosses. $A$ and $B$ are definitely independent, but if $C$ is given, then $A,B$ becomes dependent and correlated because $A+B=C$.