Large induced subgraph with a given pathwidth in outerplanar graphs

TL;DR

This paper explores the maximum size of induced subgraphs with a specified pathwidth in outerplanar graphs, extending known bounds and providing new upper bounds that generalize previous constructions.

Contribution

It extends existing results on induced linear forests to induced subgraphs with given pathwidth in outerplanar graphs, including new bounds and generalizations.

Findings

Outerplanar graphs have large induced subgraphs with bounded pathwidth.

The paper establishes upper bounds on the size of such subgraphs.

Generalizes previous results on induced linear forests to broader pathwidth conditions.

Abstract

A long-standing conjecture by Albertson and Berman states that every planar graph of order $n$ has an induced forest with at least $\lceil \frac{n}{2} \rceil$ vertices. As a variant of this conjecture, Chappell conjectured that every planar graph of order $n$ has an induced linear forest with at least $\lceil \frac{4n}{9} \rceil$ vertices. Pelsmajer proved that every outerplanar graph of order $n$ has an induced linear forest with at least $\lceil \frac{4n+2}{7}\rceil$ vertices and this bound is sharp. In this paper, we investigate the order of induced subgraphs of outerplanar graphs with a given pathwidth. The above result by Pelsmajer implies that every outerplanar graph of order $n$ has an induced subgraph with pathwidth one and at least $\lceil \frac{4n+2}{7}\rceil$ vertices. We extend this to obtain a result on the maximum order of any outerplanar graph with at most a given pathwidth. We also give its upper bound which generalizes Pelsmajer's construction.