We present a linear-time algorithm that, given a planar graph with m edges and maximum degree 3, finds an induced matching of size at least m/9. This is best possible.

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SIAM Journal on Discrete Mathematics
Networks and Optimization

Kang, R.J, Mnich, M, & Müller, T. (2012). Induced matchings in subcubic planar graphs. SIAM Journal on Discrete Mathematics, 26(3), 1383–1411.