Graphs with the Erdős-Ko-Rado property

For any graph G, we define μ(G) to be the minimum size of a maximal independent vertex set. We conjecture that, if 1⩽r⩽12μ(G), then G is r-EKR, and if r<12μ(G), then G is strictly r-EKR. This is known to be true when G is an empty graph, a cycle, a path or the disjoint union of complete graphs. We show that it is also true when G is the disjoint union of a pair of complete multipartite graphs.