On End-regular graphs Original Research Article

A graph X is End-regular if its endomorphism monoid End X is regular in the semigroup sense, that is, for any f in End X there exists g in End X with fgf = f. End-regular bipartite graphs were characterized in [1]. In this paper, End-regular graphs are obtained through the lexicographic product of End-regular graphs, and strong End-regular graphs (End-orthodox graphs) are investigated.