Planar Induced Subgraphs of Sparse Graphs

TL;DR

This paper presents new bounds on the sizes of various induced subgraphs in sparse graphs, including pseudoforests, partial 2-trees, and planar subgraphs, with constructive algorithms for their extraction.

Contribution

It provides the first linear-time algorithms for finding large induced pseudoforests, partial 2-trees, and planar subgraphs in sparse graphs, along with bounds on $K_h$-minor-free subgraphs.

Findings

Induced pseudoforest of at least n - m/4.5 vertices

Induced partial 2-tree of at least n - m/5 vertices

Induced planar subgraph of at least n - m/5.2174 vertices

Abstract

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 $n-m/6+o(m)$.