For a nonempty subset $Y$ of a nonempty set $X$, denote by $Fix(X,Y)$ the semigroup of full transformations on the set $X$ in which all elements in $Y$ are fixed. The Cayley digraph $Cay$ $(Fix(X,Y),A)$ of $Fix(X,Y)$ with respect to a connection set $A\subseteq Fix(X,Y)$ is defined as a digraph whose vertex set is $Fix(X,Y)$ and two vertices $\alpha, \beta$ are adjacent in sense of drawing a directed edge (arc) from $\alpha$ to $\beta$ if there exists $\mu\in A$ such that $\beta = \alpha\mu$. In this paper, we determine domination parameters of $Cay$ $(Fix(X,Y),A)$ where $A$ is a subset of $Fix(X,Y)$ related to minimal idempotents and permutations in $Fix(X,Y)$
Cayley digraphs of transformation semigroups, the (total/independent/connected/split) domination number
NUPO, NUTTAWOOT and POOKPIENLERT, CHOLLAWAT
"Domination parameters on Cayley digraphs of transformation semigroups with fixed sets,"
Turkish Journal of Mathematics: Vol. 45:
4, Article 23.
Available at: https://journals.tubitak.gov.tr/math/vol45/iss4/23