The inclusion ideal graph of a semigroup

In this article, we consider the inclusion ideal graph In ( S ) of nontrivial right ideals of a semigroup S with zero element. We characterize a semigroup S for which the graph In ( S ) is complete, connected and also find various graph parameters of In ( S ). We determine the values of n for which the graph In ( Z n ) is complete, triangulated, split, unicyclic, thresold and also study minimal embedding of In ( Z n ) into compact orientable (resp. non-orientable) surface. We give both upper and lower bouds for metric and partition dimension of inclusion ideal graph of a completely 0-simple semigroup. Finally, we compute some graph parameters of the cartesian product of inclusion ideal graph of two monoids.