On the design of an automaton for traversing plane labyrinths with limited holes

We study a problem on the design of an automaton that traverses all the labyrinths of a given class. We constructively design an automaton that is universal for a class of labyrinths, the diameters of whose holes are bounded by a given constant $L$. The number of states of this automaton does not exceed $L^2$ in order.