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Are there any natural decision problems which are guaranteed not to be in $\mathsf{PTIME}$? Preferably natural graph problems like $\mathsf{CLIQUE}, \mathsf{VERTEXCOVER}$ etc. (However, they would be in $\mathsf{PTIME}$ if $\mathsf{PTIME}=\mathsf{NPTIME}$).

The Time hierarchy theorem suggests a $\mathsf{EXPTIME}$ complete problem. Any graph problems which are $\mathsf{EXPTIME}$-complete?

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I am going to make manuellafond's comment to an answer:

"Cops and robbers" is a natural $\mathsf{EXPTIME}$-complete graph problem. Thus, there can be no polynomial time algorithm to decide the general "Cops and Robbers"-Problem.

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