Let Fix(X, Y) be a semigroup of full transformations on a set X in which elements in a nonempty subset Y of X are fixed. In this paper, we construct the Cayley digraphs of Fix(X, Y) and study some structural properties of such digraphs such as the connectedness and the completeness. Further, some prominent results of Cayley digraphs of Fix(X, Y) relative to minimal idempotents are verified. In addition, the characterization of an equivalence digraph of the Cayley digraph of Fix(X, Y) is also investigated.

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Let [Formula: see text] denote the Cayley digraph of the rectangular group [Formula: see text] with respect to the connection set [Formula: see text] in which the rectangular group [Formula: see text] is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. An independent dominating set of [Formula: see text] is the independent set of elements in [Formula: see text] that can dominate the whole elements. In this paper, we investigate the independent domination number of [Formula: see text] and give more results on Cayley digraphs of left groups and right groups which are specific cases of rectangular groups. Moreover, some results of the path independent domination number of [Formula: see text] are also shown.

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 ).

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