Parity of an odd dominating set

For a simple graph $G$ with vertex set $V(G)=\{v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_n\}$, we define the closed neighborhood set of a vertex $u$ as $N[u]={v \in V(G) ; | ; v ; $\text{is adjacent to}$ ; u ; $\text{or}$ ; v=u }$ and the closed neighborhood matrix $N(G)$ as the matrix obtained by setting to $1$ all the diagonal entries of the adjacency matrix of $G$. We say a set $S$ is odd dominating if $N[u]\cap S$isoddforall$u\in V(G)$. We prove that the parity of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where the rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.