Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets

Using quantum annealing to solve an optimization problem requires minor embeddings of a logic graph into a known hardware graph. In an effort to reduce the complexity of the minor embedding problem, we introduce the minor set cover (MSC) of a known graph G: a subset of graph minors which contain any remaining minor of the graph as a subgraph. Any graph that can be embedded into G will be embeddable into a member of the MSC. Focusing on embedding into the hardware graph of commercially available quantum annealers, we establish the MSC for a particular known virtual hardware, which is a complete bipartite graph. We show that the complete bipartite graph $K_{N,N}$ has a MSC of $N$ minors, from which $K_{N+1}$ is identified as the largest clique minor of $K_{N,N}$. The case of determining the largest clique minor of hardware with faults is briefly discussed but remains an open question.