On wreathed lexicographic products of graphs

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