Endomorphism Monoids of Generalized Split Graphs

Abstract

We call a graph \(G\) a \({generalized \;split \; graph}\) if there exists a core \(K\) of \(G\) such that \(V(G) \setminus V(K)\) is an independent set of \(G\).Let \(G\) be a generalized split graph with a partition \(V(G) = K \cup S\), where \(K\) is a core of \(G\) and \(S\) is an independent set. We prove that \(G\) is end-regular if and only if for any \(a, b \in S\), \(\phi \in \text{Aut}(K)\), the inclusion \(\phi(N(a)) \subsetneqq N(b)\) does not hold.
\(G\) is end-orthodox if and only if \(G\) is end-regular and for any \(a, b \in S\), \(N(a) \neq N(b)\).