Let the group $G$ have the presentation $\langle x_1, \dots, x_n \;|\; r_1, \dots, r_m \rangle$. Let $\Gamma$ be a labelled  directed graph corresponding to Van Kampen diagram over the above presentation of $G$. Let $\Gamma_1$ be the undirected graph corresponding to $\Gamma$. 

Let $\mathcal{C}$ be an arbitrary set of cycles of $\Gamma_1$ such that each edge of $\Gamma_1$ belongs to at least one of the edges of the cycles of $\mathcal{C}$. Let $R_1, \dots,  R_k$ be the labels of  cycles in $\mathcal{C}$, (where by the label of a cycle it is meant the word read on the cycle.)

Is it true that $\langle x_1, \dots, x_n \;|\; R_1, \dots, R_k \rangle$ is a presentation for $G$?  

If $\mathcal{C}$ generates the cycle space of $\Gamma_1$, then the answer is positive.