Date of Original Version
We investigate the computational complexity of deciding whether k cops can capture a robber on a graph G. Goldstein and Reingold (1995)  conjectured that the problem is EXPTIME-complete when both G and k are part of the input; we prove this conjecture.
Kinnersley, W. B. (2014). Cops and Robbers is EXPTIME-complete. Journal of Combinatorial Theory, Series B, 2011, 201-220.
Available at http://dx.doi.org/10.1016/j.jctb.2014.11.002.