@article{Nupo2024, 
author = {Nuttawoot Nupo and Chollawat Pookpienlert},
title = {Fractional domination and fractional total domination on Cayley digraphs of transformation semigroups with fixed sets},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {6},
pages = {14558-14573},
keywords = {Cayley digraph, transformation semigroup with fixed set, fractional dominating function, fractional total dominating function, fractional domination number, fractional total domination number},
url = {https://www.sciopen.com/article/10.3934/math.2024708},
doi = {10.3934/math.2024708},
abstract = {For a set    X and a nonempty subset    Y of    X, denote by    T  (  X  ) the full transformation semigroup under the composition whose elements are functions on    X. Let    F  i  x  (  X  ,  Y  ) be the subsemigroup of    T  (  X  ) containing functions    α  ∈  T  (  X  ) in which each element in    Y is a fixed point of    α. Moreover, let    A be a nonempty subset of    F  i  x  (  X  ,  Y  ). The Cayley digraph of    F  i  x  (  X  ,  Y  ) with respect to a connection set    A is a digraph with vertex set    F  i  x  (  X  ,  Y  ) and two vertices    α  ,  β induce an arc    (  α  ,  β  ) if    β  =  α  λ for some    λ  ∈  A. In this paper, the concepts of fractional dominating and fractional total dominating functions of those Cayley digraphs were investigated. Furthermore, the fractional domination and fractional total domination numbers were determined.}
}