Given a finite simple graph $\Gamma$, the graph group $G \Gamma$ is the group with generators in one-to-one correspondence with the vertices of $\Gamma$ and with relations in one-to-one correspondence with the edges of $\Gamma$: two generators commute if and only if their associated vertices are adjacent in $\Gamma$. Let $\phi \colon G \Gamma \to \mathbb Z$ be a homomorphism which maps each generator to 0 or 1. We derive an explicit presentation for $\ker \phi$, and give a condition, dependent on $\Gamma$ and $\phi$, which guarantees the finite presentation of $\ker \phi$.
"Finite Presentations of Subgroups of Graph Groups." Missouri J. Math. Sci. 10 (2) 70 - 82, Spring 1998. https://doi.org/10.35834/1998/1002070