The problem of determining whether a given graph is a core graph is easily seen to be in co-NP. In fact, it is co-NP complete.
The problem of determining whether a given subgraph H is a core of a given graph G is in the larger class DP (https://complexityzoo.uwaterloo.ca/Complexity_Zoo:D#dp), and is in fact complete for this class (the archetypical complete problem for this class consists of pairs of boolean formulas, where the first one is satisfiable and the second one is unsatifiable). Containment in DP is clear: test that G maps homomorphically to H (this is encoded as satisfiability) and simultaneously that H has no homomorphism to itself which is not onto (this is encoded as unsatisfiability). DP-hardness is nontrivial, and is proved in the paper:
Fagin, Ronald, Phokion G. Kolaitis, and Lucian Popa. "Data exchange: getting to the core." ACM Transactions on Database Systems (TODS) 30.1 (2005): 174-210.