  In the {\sc Cluster Deletion} problem the goal is to remove the minimum
number of edges of a given graph, such that every connected component of the
resulting graph constitutes a clique. It is known that the decision version of
{\sc Cluster Deletion} is NP-complete on ($P_5$-free) chordal graphs, whereas
{\sc Cluster Deletion} is solved in polynomial time on split graphs. However,
the existence of a polynomial-time algorithm of {\sc Cluster Deletion} on
interval graphs, a proper subclass of chordal graphs, remained a well-known
open problem. Our main contribution is that we settle this problem in the
affirmative, by providing a polynomial-time algorithm for {\sc Cluster
Deletion} on interval graphs. Moreover, despite the simple formulation of the
algorithm on split graphs, we show that {\sc Cluster Deletion} remains
NP-complete on a natural and slight generalization of split graphs that
constitutes a proper subclass of $P_5$-free chordal graphs. To complement our
results, we provide two polynomial-time algorithms for {\sc Cluster Deletion}
on subclasses of such generalizations of split graphs.
