Černý conjecture for edge-colored digraphs with few junctions

In this paper we consider the Černý conjecture in terminology of colored digraphs corresponding to finite automata. We define a class of colored digraphs having a relatively small number of junctions between paths determined by different colors, and prove that digraphs in this class satisfy the Černý conjecture. We argue that this yields not only a new class of automata for which the Černý conjecture is verified, but also that our approach may be viewed as a new more systematic way to attack the Černý conjecture in its generality, giving an insight into the complexity of the problem.