Planar Induced Subgraphs of Sparse Graphs

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 K h -minor-free graph in a given graph can sometimes be at most n − m/6 + o(m)