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Since planar triangle-free graphs are 3-colourable, such a graph with nvertices has an independent set of size at least n/3. We prove that unless thegraph contains a certain obstruction, its independence number is at leastn/(3-epsilon) for some fixed epsilon>0. We also provide a reduction rule forthis obstruction, which enables us to transform any plane triangle-free graph Ginto a plane triangle-free graph G' such that alpha(G')-|G'|/3=alpha(G)-|G|/3and |G'|<=(alpha(G)-|G|/3)/epsilon. We derive a number of algorithmicconsequences as well as a structural description of n-vertex planetriangle-free graphs whose independence number is close to n/3.

Triangle-free planar graphs with small independence number