Edit Tony's answer is quite nice, but I meant something else. Sorry for editing again, I meant edges.

I am looking for a graph with 3 distinguished edges $xx'$,$yy'$,$zz'$ where $\deg(x)=\deg(y)=\deg(z)=1$.

One can chose arbitrary weights for the edges and the graph must satisfy:

1. Must have at least two maximum weighted matchings in one of which all of the 3 distinguished edges are present and in the other all are not present.
2. For all maximum weighted matchings (if more than 2) the distinguished edges are either all present or all not present.

Need this for a graph gadget and suspect it is quite unlikely to exist.

For only 2 distinguished edges a trivial solution is the path with 3 edges $v v' v'' v'''$.