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Every $n$-vertex planar triangle-free graph with maximum degree at most $3$has an independent set of size at least $\frac{3}{8}n$. This was firstconjectured by Albertson, Bollob\'as and Tucker, and was later proved byHeckman and Thomas. Fraughnaugh and Locke conjectured that the planarityrequirement could be relaxed into just forbidding a few specific nonplanarsubgraphs: They described a family $\mathcal{F}$ of six nonplanar graphs (eachof order at most $22$) and conjectured that every $n$-vertex triangle-freegraph with maximum degree at most $3$ having no subgraph isomorphic to a memberof $\mathcal{F}$ has an independent set of size at least $\frac{3}{8}n$. Inthis paper, we prove this conjecture.  As a corollary, we obtain that every $2$-connected $n$-vertex triangle-freegraph with maximum degree at most $3$ has an independent set of size at least$\frac{3}{8}n$, with the exception of the six graphs in $\mathcal{F}$. Thisconfirms a conjecture made independently by Bajnok and Brinkmann, and byFraughnaugh and Locke.

Large independent sets in triangle-free cubic graphs: beyond planarity