Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

Abstract

For a nonempty subset Y of a nonempty set X , denote by F ix(X, Y ) the semigroup of full transformations on the set X in which all elements in Y are fixed. The Cayley digraph Cay(F ix(X, Y ), A) of F ix(X, Y ) with respect to a connection set A ⊆ F ix(X, Y ) is defined as a digraph whose vertex set is F ix(X, Y ) and two vertices α, β are adjacent in sense of drawing a directed edge (arc) from α to β if there exists µ ∈ A such that β = αµ. In this paper, we determine domination parameters of Cay(F ix(X, Y ), A) where A is a subset of F ix(X, Y ) related to minimal idempotents and permutations in F ix(X, Y ).