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 embedding 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 \({\mathcal {G}}\): a subset of graph minors which contain any remaining minor of the graph as a subgraph. Any graph that can be embedded into \({\mathcal {G}}\) 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.