Very well-covered graphs with the Erd?s-Ko-Rado property

A family of independent r-sets of a graph G is an r-star if every set in the family contains some fixed vertex v. A graph is r-EKR if the maximum size of an intersecting family of independent r-sets is the size of an r-star. Holroyd and Talbot conjectured that a graph is r-EKR as long as 1 < r < 21 mu(G), where mu(G) is the minimum size of a maximal independent set. It is suspected that the smallest counterexample to this conjecture is a well-covered graph. Here we consider the class of very well-covered graphs G obtained by appending a single pendant edge to each vertex of G. We prove that the pendant complete graph Kn is r-EKR when n > 2r and strictly so when n > 2r. Pendant path graphs Pn are also explored and the vertex whose r-star is of maximum size is determined.