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）, 1End（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 1End（X） = qEnd（X） for any split graph X. The conditions under which 1End（X） （resp. qEnd（X）） forms a submonoid of End（X） are also given. In particular, if hEnd（X） forms a monoid, then 1End（X） （resp. qEnd（X）） forms a monoid too.
In this paper, the half-strong endomorphisms of the join of split graphs are investigated. We give the conditions under which the half-strong endomorphisms of the join of split graphs form a monoid.
