Improved bounds on the cop number when forbidding a minor

Abstract Andreae proved that the cop number of connected ‐minor‐free graphs is bounded for every graph . In particular, the cop number is at most if contains no isolated vertex, where . The main result of this paper is an improvement on this bound, which is most significant when is small or sparse, for instance, when 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 ‐minor‐free graphs, ‐minor‐free graphs and linklessly embeddable graphs.