Triangle-free planar graphs with small independence number

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Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n ∕ 3 . We prove that unless the graph contains a certain obstruction, its independence number is at least n ∕ ( 3 − ε ) for some fixed ε > 0 . We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-free graph G ′ such that α ( G ′ ) − | G ′ | ∕ 3 = α ( G ) − | G | ∕ 3 and | G ′ | ≤ ( α ( G ) − | G | ∕ 3 ) ∕ ε . We derive a number of algorithmic consequences as well as a structural description of n -vertex plane triangle-free graphs whose independence number is close to n ∕ 3 .