I don’t think that works for the self recursion trick. It’s a logical problem more than a representation problem. Even if you had a machine to decide halting status quickly, it’d be broken; representing computation as a graph can’t help.

The inputs to a halting decider TM is an infinite set (all TMs and all inputs to them) so that’s another reason why a finite graph can’t be represented. Even something like integer addition isn’t a finite graph if you enumerate all inputs