Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph

TL;DR

This paper proves that 2-outerplanar graphs contain large induced subgraphs that are either acyclic or outerplanar, improving bounds known for general planar graphs and providing tighter results for this specific class.

Contribution

It establishes that every 2-outerplanar graph has an induced forest covering at least half of its vertices and an induced outerplanar subgraph covering at least two-thirds, under certain conditions.

Findings

Induced forest on at least half the vertices in 2-outerplanar graphs

Induced outerplanar subgraph on at least two-thirds of vertices with two-connected inner layers

Tighter bounds for specific subclasses of planar graphs

Abstract

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for 2-outerplanar graphs. We show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest. We also show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices, assuming that the connected components of the inner layer are two-connected.