  One of the more recent generalizations of the Erd\"os-Ko-Rado theorem,
formulated by Holroyd, Spencer and Talbot, defines the Erd\"os-Ko-Rado property
for graphs in the following manner: for a graph G and a positive integer r, G
is said to be r-EKR if no intersecting subfamily of the family of all
independent vertex sets of size r is larger than the largest star, where a star
centered at a vertex v is the family of all independent sets of size $r$
containing v. In this paper, we prove that if G is a disjoint union of chordal
graphs, including at least one singleton, then G is r-EKR if $r\leq mu(G)/2$,
where mu(G) is the minimum size of a maximal independent set. We will also
prove Erd\"os-Ko-Rado results for chains of complete graphs, which are a class
of chordal graphs obtained by blowing up edges of a path into complete graphs.
We also consider similar problems for ladder graphs and trees, and prove
preliminary results for these graphs.
