What's the complexity of counting odd nodes in graph?
Well, at least $\#\mathsf{P}$-hard. Given a SAT formula, construct a graph with two vertices, $v_x$ and $v_x'$, for every possible assignment of variables $\vec{x}$. If $x$ is a satisfying assignment for the formula, draw an edge between $v_x$ and $v_x'$; these are the only edges. It is easy to construct the circuit for this graph from the SAT formula, and the number of odd vertices is exactly twice the number of satisfying assignments.