On wreathed lexicographic products of graphs

This paper proves a necessary and sufficient condition for the endomorphism monoid E n d G [ H ] of a lexicographic product G [ H ] of graphs G , H to be the wreath product E n d G ≀ E n d H of the monoids E n d G and E n d H . The paper also gives respective necessary and sufficient conditions for specialized cases such as for unretractive or triangle-free graphs G .