A family of independent
-sets
of a graph
is an
-star
if every set in the family contains some fixed vertex
. A graph
is
-EKR
if the maximum size of an intersecting family of independent
-sets is the size
of an
-star.
Holroyd and Talbot conjectured that a graph is
-EKR as
long as
,
where
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
obtained by appending a single pendant edge to each vertex of
. We prove that the
pendant complete graph
is
-EKR when
and strictly so when
. Pendant path graphs
are also explored and
the vertex whose
-star
is of maximum size is determined.