Domination parameters on Cayley digraphs of full transformation semigroups relative to Green’s equivalence LL-classes and their complements

Let T ( X ) be a full transformation semigroup on a nonempty set X . The Cayley digraph Γ of a semigroup T ( X ) with respect to the Green’s <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">L</m:mi> </m:math> L -class is a digraph with vertex set T ( X ) and two vertices α , β ∈ T ( X ) are adjacent as an arc ( α , β ) whenever β = αμ for some μ in a connection set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">L</m:mi> </m:math> L -class. In this paper, we investigate the domination parameters of Cayley digraphs Γ and their complements <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> <m:mo>̄</m:mo> </m:mover> </m:mrow> </m:math> Γ consisting of domination, total domination, independent domination, connected domination, and split domination.