Max independent set in planar graphs PTAS proof
I haven't got any paper, but looking through my favorite book regarding approximation algorithms i've got something.

It is named: The Design of Approximation Algorithms.
You find it free for download (even legal) on https://www.designofapproxalgs.com.

Take a look at page 269 (ff.) especially page 271 should be interesting for you as they describe and proof a PTAS for max independent set in planar graphs.

It might not be "front of the line" research or results but hope it helps.