Planar Induced Subgraphs of Sparse Graphs

We show that every graph has an induced pseudoforest of at least $n-m\mathrm{/}4.5$ vertices, an induced partial 2-tree of at least $n-m\mathrm{/}5$ vertices, and an induced planar subgraph of at least $n-m\mathrm{/}5.2174$ 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 $n-m\mathrm{/}6+o(m)$.