Will a vertex present in the set of vertices present in each basis cycle be present in all the cycles? I read that the XOR of basis cycles can generate all the cycles present in the graph.

I was just wondering if there exists any one vertex which is present in all the basis cycles and is present in all the cycles of the graph.

Can someone point me to something that might help me tackle this problem?
Consider the following graph:



Let $S$ be the set containing the cycles formed by i) the red and black edges, and ii) the blue and black edges. $S$ is a cycle basis in which every cycle contains vertex $x$, yet the cycle formed by the red and blue edges does not contain $x$. 



Edit: it is also not true that the existence of a vertex $x$ that lies in all cycles of a cycle basis implies that $x$ belongs to all cycles of the graph. See the following counterexample:



The $3$ cycles that can be formed by taking an edge $e$ of color $c$ for $c \in \{ \mbox{red, green, blue}\}$ along with the two black edges incident to the endpoints of $e$ form a basis $S$. Vertex $x$ belongs to all the cycles in $S$ but it does not belong to the cycle formed by the red, green, and blue edges.