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, where $h\in V(H)$. 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 $K_{3,t}$-minor-free graphs, $K_{2,t}$-minor-free graphs and linklessly embeddable graphs.

翻译：Andreae (1986) 证明了对于任意图H，连通H-子式禁止图的警察数是有界的。特别地，若H-h不含孤立顶点（其中h∈V(H)），则警察数至多为|E(H-h)|。本文的主要结果是对该界的改进，当H较小或稀疏时（例如当H-h可通过另一图的多重边细分得到时）改进尤为显著。该结果的部分推论包括对K_{3,t}-子式禁止图、K_{2,t}-子式禁止图以及无链环嵌入图警察数上界的改进。