# Improved bounds on the cop number when forbidding a minor

@article{Kenter2023ImprovedBO, title={Improved bounds on the cop number when forbidding a minor}, author={Franklin Kenter and Erin Meger and J{\'e}r{\'e}mie Turcotte}, journal={ArXiv}, year={2023}, volume={abs/2302.14851} }

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…

## 23 References

### Bounding the Cop Number of a Graph by Its Genus

- 2021

Mathematics

SIAM J. Discret. Math.

The first improvement to Schroder's bound on cop number of a connected graph is given, showing that c(G) is bounded as a function of the genus of the graph g(G).

### Cops and Robbers on Graphs with a Set of Forbidden Induced Subgraphs

- 2020

Mathematics

Theor. Comput. Sci.

### Topological directions in Cops and Robbers

- 2017

Mathematics

ArXiv

This work surveys results at the intersection of topological graph theory and the game of Cops and Robbers, focusing on results, conjectures, and open problems for the cop number of a graph embedded on a surface, and considers graphs of higher genus.

### The Cops and Robber game on graphs with forbidden (induced) subgraphs

- 2010

Mathematics

Contributions Discret. Math.

This paper completely characterize (for both relations) the graphs which force bounded cop number in the classes of graphs defined by forbidding one or more graphs as either sub graphs or induced subgraphs and bound theCop number in terms of the tree-width.

### On a pursuit game played on graphs for which a minor is excluded

- 1986

Mathematics

J. Comb. Theory, Ser. B

### Cops and robbers on P5-free graphs

- 2023

Mathematics

ArXiv

We prove that every connected P 5 -free graph has cop number at most two, solving a conjecture of Sivaraman. In order to do so, we ﬁrst prove that every connected P 5 -free graph G with independence…

### The Colin de Verdière graph parameter

- 1999

Mathematics

In 1990, Y. Colin de Verdi ere introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity…

### Linkless embeddings of graphs in 3-space

- 1993

Mathematics, Computer Science

Any two flat embeddings of a 4-connected graph are either ambient isotopic, or one is ambient isotopy to a mirror image of the other.