Large induced subgraph with a given pathwidth in outerplanar graphs
Naoki Matsumoto, Takamasa Yashima
TL;DR
The paper addresses how large an induced subgraph of an outerplanar graph can be while preserving pathwidth at most $k$, formalizing the problem via $I_k(G)$ and $M_k$. It proves a general lower bound $I_k(G) \ge \frac{M_k}{M_k+3}n$, with a sharp bound for $k=2$ yielding $I_2(G) \ge \frac{5n}{7}$ since $M_2=5$, and provides a near-tight upper bound by a constructive extremal graph, highlighting tightness evidence. The results extend the line of study on induced forests/linear forests to subgraphs with bounded pathwidth in outerplanar graphs, connecting to prior work by Chappell and Pelsmajer. The work also outlines implications for planar graphs and poses open problems about $I_k(G)$ in the planar setting, offering a framework for future exploration of induced subgraphs with constrained pathwidth in broader graph classes.
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.