Graphs with the Erdős–Ko–Rado property

AbstractFor a graph G vertex v of G and integer r⩾1, we denote the family of independent r-sets of V(G) by I(r)(G) and the subfamily {A∈I(r)(G):v∈A} by Iv(r)(G); such a subfamily is called a star. Then, G is said to be r-EKR if no intersecting subfamily of I(r)(G) is larger than the largest star in I(r)(G). If every intersecting subfamily of Iv(r)(G) of maximum size is a star, then G is said to be strictly r-EKR. We show that if a graph G is r-EKR then its lexicographic product with any complete graph is r-EKR.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