A long-standing conjecture by Albertson and Berman in 1979 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. As a partial solution to the conjecture, Pelsmajer in 2004 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 with a given pathwidth in outerplanar graphs. The above result of Pelsmajer implies that every outerplanar graph of order $n$ has an induced subgraph with pathwidth at most 1 and at least $\lceil \frac{4n+2}{7}\rceil$ vertices. We extend this to obtain a result on the maximum order of induced subgraphs with a given pathwidth in an outerplanar graph. We also give its upper bound, which generalizes Pelsmajer's construction.

翻译：[摘要] 1979年Albertson与Berman提出的长期未解猜想指出：每个$n$阶平面图均包含至少$\lceil \frac{n}{2} \rceil$个顶点的诱导森林。作为该猜想的变体，Chappell推测每个$n$阶平面图均存在至少$\lceil \frac{4n}{9} \rceil$个顶点的诱导线性森林。2004年，Pelsmajer部分证明了该猜想，证得每个$n$阶外平面图均包含至少$\lceil \frac{4n+2}{7}\rceil$个顶点的诱导线性森林，且该界是紧的。本文研究外平面图中给定路径宽度的诱导子图阶数。上述Pelsmajer结果蕴含每个$n$阶外平面图均存在路径宽度至多为1且顶点数至少为$\lceil \frac{4n+2}{7}\rceil$的诱导子图。我们推广这一结论，得到了外平面图中给定路径宽度诱导子图的最大阶数结果，并给出了其上限界，该上界推广了Pelsmajer的构造。