Andreae (1986) proved that the cop number of connected $H$-minor-free graphs is bounded for every graph $H$. In particular, the cop number is at most $|E(H-h)|$ if $H-h$ contains no isolated vertex. The main result of this paper is an improvement on this bound, which is most significant when $H$ is small or sparse, for instance when $H-h$ can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of $K_{3,m}$-minor-free graphs, $K_{2,m}$-minor free graphs and linklessly embeddable graphs.

翻译：Andreae (1986) 证明了对于任意图 $H$，连通且不含 $H$ 作为子式的图类中的警察数有界。具体而言，若 $H-h$ 不含孤立顶点，则警察数至多为 $|E(H-h)|$。本文的主要成果是对该上界的改进，当 $H$ 较小或稀疏时（例如 $H-h$ 可由另一图通过多次边细分得到）改进尤为显著。该结果可推得 $K_{3,m}$-子式自由图、$K_{2,m}$-子式自由图以及不可链式嵌入图的警察数上界改进。