Planar Induced Subgraphs of Sparse Graphs - Borradaile, Glencora; Eppstein, David; Zhu, Pingan (Journal of Graph Algorithms and Applications, 2015)
We show that every graph has an induced pseudoforest of at least n−m/4.5 vertices, an induced partial 2-tree of at least n−m/5 vertices, and an induced planar subgraph of at least n−m/5.2174 vertices. These results are constructive, implying linear-time algorithms to find the respective induced subgraphs. We also show that the size of the largest Kh-minor-free graph in a given graph can sometimes be at most n−m/6+o(m).