Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph
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.
Keywords
Cite
@article{arxiv.1711.00212,
title = {Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph},
author = {Glencora Borradaile and Hung Le and Melissa Sherman-Bennett},
journal= {arXiv preprint arXiv:1711.00212},
year = {2026}
}
Comments
13 pages, 7 figures. Accepted to Graphs and Combinatorics. v2: Added additional assumption to Theorem 15 (see Remark 16). We thank D'Elia and Frati for pointing out the necessity of this assumption to our argument