Cayley Graphs of Transformation Semigroups with Restriction on the Fixed Set is Bijective (European Journal of Pure and Applied Mathematics, 2026)
Let $PG_Y(X)=\{α \in T(X) : α|_Y \in G(Y)\}$ be a subsemigroup of the full transformation semigroup $T(X)$, where the restriction of each transformation to $Y$ is a permutation of $Y$. This structure can be viewed as an extension of the symmetric group to a transformation semigroup.In this paper, we study the Cayley digraphs $\operatorname{Cay}(PG_Y(X),A)$ generated by a nonempty set $A$ of minimal idempotents. We describe basic structural properties of these digraphs, including initial and terminal vertices. We also examine their degrees, connectedness, and completeness. In addition, domination parameters of these digraphs are determined.
Keywords: transformation semigroup; Cayley graph; connectedness; domination parameters