  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 conjecture that a graph is $r$-EKR as long as
$1\leq r\leq\frac{\mu(G)}{2}$, 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 $K_n^*$ is $r$-EKR when
$n \geq 2r$ and strictly so when $n>2r$. Pendant path graphs $P_n^*$ are also
explored and the vertex whose $r$-star is of maximum size is determined.
