In this paper, we introduce and study the following question. Let $\mathcal G$ be a family of graphs and let $k\geq 3$ be an integer. What is the largest value $f_k(n)$ such that every $n$-vertex graph in $\mathcal G$ has an induced subgraph with degree at most $k$ and with $f_k(n)$ vertices? Similar questions, in which one seeks a large induced forest, or a large induced linear forest, or a large induced $d$-degenerate graph, rather than a large induced graph of bounded degree, have been studied for decades and have given rise to some of the most fascinating and elusive conjectures in Graph Theory. We tackle our problem when $\mathcal G$ is the class of the outerplanar graphs, or the class of the planar graphs, or the class of the graphs whose degree is bounded by a value $d>k$. In all cases, we provide upper and lower bounds on the value of $f_k(n)$. For example, we prove that every $n$-vertex planar graph has an induced subgraph with degree at most $3$ and with $\frac{5n}{13}>0.384n$ vertices, and that there exist $n$-vertex planar graphs whose largest induced subgraph with degree at most $3$ has $\frac{4n}{7}+O(1)<0.572n+O(1)$ vertices.

翻译：本文引入并研究以下问题：设$\mathcal G$为一个图族，$k\geq 3$为整数。对于$\mathcal G$中任意$n$个顶点的图，其存在一个度至多为$k$且顶点数为$f_k(n)$的诱导子图，则$f_k(n)$的最大值是多少？数十年来，研究者们一直在探讨类似问题——寻找大型诱导森林、大型诱导线性森林或大型诱导$d$-退化图，而非有界度的大型诱导图，这些问题催生了图论中一些最引人入胜且难以捉摸的猜想。本文针对$\mathcal G$为外平面图类、平面图类或最大度受限于某值$d>k$的图类的情形展开研究。在所有情况下，我们均给出了$f_k(n)$值的上界与下界。例如，我们证明每个$n$顶点平面图都存在一个度至多为$3$、顶点数至少为$\frac{5n}{13}>0.384n$的诱导子图；同时存在一类$n$顶点平面图，其度至多为$3$的最大诱导子图的顶点数不超过$\frac{4n}{7}+O(1)<0.572n+O(1)$。