  In this paper, we give a characterization of digraphs $Q, |Q|\leq 4$ such
that the associated Hecke-Kiselman monoid $H_Q$ is finite. In general, a
necessary condition for $H_Q$ to be a finite monoid is that $Q$ is acyclic and
its Coxeter components are Dynkin diagram. We show, by constructing examples,
that such conditions are not sufficient.
