I am looking for a proof why the minor embedding problem (or minor graph search or minor testing) belongs to NP-complete problems. I would like to find a paper or in general an explanation that shows the relation to other NP-problems, e.g. how to reduce the search for the minor graph to another NP-complete problem.
Looking online I have found many articles that show specific cases of the problem that can be solved polynomially, but that is not what I am looking for.
Proof need to be done case by case to different graph structures to which minor embedding is done.
To structures that are relevant to D-Wave machine: "Minor embedding in broken chimera and derived graphs is NP-complete" https://www.sciencedirect.com/science/article/pii/S0304397523006825
In this work, we show the hardness of the embedding problem for all types of currently existing hardware, represented by the Chimera and the derived Pegasus and Zephyr graphs, containing unavailable qubits. We construct certain broken Chimera graphs, where it is hard to find a Hamiltonian cycle. As the Hamiltonian cycle problem is a special case of the embedding problem, this proves the general complexity result for the Chimera graphs. By exploiting the subgraph relation between the Chimera and the derived graphs, the proof is then further extended to the Pegasus graphs and to the Zephyr graphs.
