Motivated by a conjecture from the automated conjecturing program TxGraffiti, in this paper the relationship between the zero forcing number, $Z(G)$, and the vertex independence number, $α(G)$, of cubic and subcubic graphs is explored. TxGraffiti conjectures that for all connected cubic graphs $G$, that are not $K_4$, $Z(G) \leq α(G) + 1$. This work uses decycling partitions of upper-embeddable graphs to show that almost all cubic graphs satisfy $Z(G) \leq α(G) + 2$, provides an infinite family of cubic graphs where $Z(G) = α(G) + 1$, and extends known bounds to subcubic graphs.