Split graphs whose half-strong endomorphisms form a monoid

In this paper, the half-strong, the locally strong and the quasi-strong endomorphisms of a split graph are investigated. Let X be a split graph and let End(X), hEnd(X), lEnd(X) and qEnd(X) be the endomorphism monoid, the set of all half-strong endomorphisms, the set of all locally strong endomorphisms and the set of all quasi-strong endomorphisms of X, respectively. The conditions under which hEnd(X) forms a submonoid of End(X) are given. It is shown that lEnd(X) = qEnd(X) for any split graph X. The conditions under which lEnd(X) (resp. qEnd(X)) forms a submonoid of End(X) are also given. In particular, if hEnd(X) forms a monoid, then lEnd(X) (resp. qEnd(X)) forms a monoid too.