End-Completely-Regular and End-Inverse Lexicographic Products of Graphs

Abstract

A graphXis said to be End-completely-regular (resp., End-inverse) if its endomorphism monoid End(X) is completely regular (resp., inverse). In this paper, we will show that ifX[Y] is End-completely-regular (resp., End-inverse), then bothXandYare End-completely-regular (resp., End-inverse). We give several approaches to construct new End-completely-regular graphs by means of the lexicographic products of two graphs with certain conditions. In particular, we determine the End-completely-regular and End-inverse lexicographic products of bipartite graphs.