End-Completely-Regular and End-Inverse Joins of Graphs

Abstract

A graph \(X\) is said to be \({End-completely-regular}\) (\({End-inverse}\)) if its endomorphism monoid \(End(X)\) is completely regular (inverse). In this paper, we demonstrate that if \(X + Y\) is End-completely-regular, then both \(X\) and \(Y\) are End-completely-regular. We present several approaches to construct new End-completely-regular graphs via the join of two graphs with specific conditions. Notably, we determine the End-completely-regular joins of bipartite graphs. Furthermore, we prove that \(X + Y\) is End-inverse if and only if \(X + Y\) is End-regular and both \(X\) and \(Y\) are End-inverse. Additionally, we determine the End-inverse joins of bipartite graphs.