Stackings and the W-cycle conjecture

We prove Wises $W$-cycles conjecture. Consider a compact graph $Gamma$ immering into another graph $Gamma$. For any immersed cycle $Lambda:S^1to Gamma$, we consider the map $Lambda$ from the circular components $mathbb{S}$ of the pullback to $Gamma$. Unless $Lambda$ is reducible, the degree of the covering map $mathbb{S}to S^1$ is bounded above by minus the Euler characteristic of $Gamma$. As a consequence, we obtain a homological version of coherence for one-relator groups.