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 $ Fix(X, Y) $ be the subsemigroup of $ T(X) $ containing functions $ \alpha\in T(X) $ in which each element in $ Y $ is a fixed point of $ \alpha $. Moreover, let $ A $ be a nonempty subset of $ Fix(X, Y) $. The Cayley digraph of $ Fix(X, Y) $ with respect to a connection set $ A $ is a digraph with vertex set $ Fix(X, Y) $ and two vertices $ \alpha, \beta $ induce an arc $ (\alpha, \beta) $ if $ \beta = \alpha\lambda $ for some $ \lambda\in 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.
Citation: Nuttawoot Nupo, Chollawat Pookpienlert. Fractional domination and fractional total domination on Cayley digraphs of transformation semigroups with fixed sets[J]. AIMS Mathematics, 2024, 9(6): 14558-14573. doi: 10.3934/math.2024708
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 $ Fix(X, Y) $ be the subsemigroup of $ T(X) $ containing functions $ \alpha\in T(X) $ in which each element in $ Y $ is a fixed point of $ \alpha $. Moreover, let $ A $ be a nonempty subset of $ Fix(X, Y) $. The Cayley digraph of $ Fix(X, Y) $ with respect to a connection set $ A $ is a digraph with vertex set $ Fix(X, Y) $ and two vertices $ \alpha, \beta $ induce an arc $ (\alpha, \beta) $ if $ \beta = \alpha\lambda $ for some $ \lambda\in 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.
| [1] |
S. Arumugam, V. Mathew, K. Karuppasamy, Fractional distance domination in graphs, Discuss. Math. Graph T., 32 (2012), 449–459. http://doi.org/10.7151/dmgt.1609 doi: 10.7151/dmgt.1609
|
| [2] | S. Arumugam, K. Rejikumar, Fractional independence and fractional domination chain in graphs, AKCE Int. J. Graphs Co., 4 (2007), 161–169. |
| [3] | A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, USA: American Mathematical Society, 1961. |
| [4] |
D. C. Fisher, Domination, fractional domination, 2-packing, and graph products, SIAM J. Discrete Math., 7 (1994), 493–498. http://doi.org/10.1137/S0895480191217806 doi: 10.1137/S0895480191217806
|
| [5] |
D. C. Fisher, Fractional dominations and fractional total dominations of graph complements, Discrete Appl. Math., 122 (2002), 283–291. http://doi.org/10.1016/S0166-218X(01)00305-5 doi: 10.1016/S0166-218X(01)00305-5
|
| [6] |
D. C. Fisher, J. Ryan, G. Domke, A. Majumdar, Fractional domination of strong direct products, Discrete Appl. Math., 50 (1994), 89–91. http://doi.org/10.1016/0166-218X(94)90165-1 doi: 10.1016/0166-218X(94)90165-1
|
| [7] | G. H. Fricke, E. O. Hare, D. P. Jacobs, A. Majumdar, On integral and fractional total domination, Congr. Numer., 77 (1990), 87–95. |
| [8] | E. O. Hare, Fibonacci numbers and fractional domination of $P_m\times P_n$, Fibonacci Quart., 32 (1994), 69–73. |
| [9] |
A. Harutyunyan, T. N. Le, A. Newman, S. Thomassé, Domination and fractional domination in digraphs, Electron. J. Comb., 25 (2018). http://doi.org/10.37236/7211 doi: 10.37236/7211
|
| [10] | S. M. Hedetniemi, S. T. Hedetniemi, T. V. Wimer, Linear time resource allocation algorithms for trees, In Southeastern Conference on Combinatorics, Graph Theory and Computing, 1987. |
| [11] |
P. Honyam, J. Sanwong, Semigroups of transformations with fixed sets, Quaest. Math., 36 (2013), 79–92. http://doi.org/10.2989/16073606.2013.779958 doi: 10.2989/16073606.2013.779958
|
| [12] | J. M. Howie, Fundamentals of semigroup theory, New York: Oxford University Press, 1995. |
| [13] |
M. G. Karunambigai, A. Sathishkumar, Dominating function in fractional graph, J. Phys. Conf. Ser., 1947 (2021). http://doi.org/10.1088/1742-6596/1947/1/012014 doi: 10.1088/1742-6596/1947/1/012014
|
| [14] |
N. Nupo, C. Pookpienlert, Domination parameters on Cayley digraphs of transformation semigroups with fixed sets, Turk. J. Math., 45 (2021), 1775–1788. http://doi.org/10.3906/mat-2104-18 doi: 10.3906/mat-2104-18
|
| [15] |
N. Nupo, C. Pookpienlert, On connectedness and completeness of Cayley digraphs of transformation semigroups with fixed sets, Int. Electron. J. Algeb., 28 (2020), 110–126. http://doi.org/10.24330/ieja.768190 doi: 10.24330/ieja.768190
|
| [16] | S. Pirzada, An introduction to graph theory, Hyderabad: Universities Press, 2012. |
| [17] |
R. Rubalcaba, M. Walsh, Minimum fractional dominating functions and maximum fractional packing functions, Discrete Math., 309 (2009), 3280–3291. http://doi.org/10.1016/j.disc.2008.09.049 doi: 10.1016/j.disc.2008.09.049
|
| [18] | K. J. Sangeetha, B. Maheswari, Basic minimal dominating functions of Euler Totient Cayley graphs, IOSR J. Math., 11 (2015), 50–58. |
| [19] | E. R. Scheinerman, D. H. Ullman, Fractional graph theory: A rational approach to the theory of graphs, New York: John Wiley $ & $ Sons, 1997. |
| [20] |
P. Shanthi, S. Amutha, N. Anbazhagan, S. B. Prabu, Effects on fractional domination in graphs, J. Intell. Fuzzy Syst., 44 (2023), 7855–7864. http://doi.org/10.3233/JIFS-222999 doi: 10.3233/JIFS-222999
|
| [21] |
M. Walsh, Fractional domination in prisms, Discuss. Math. Graph T., 27 (2007), 541–547. http://doi.org/10.7151/dmgt.1378 doi: 10.7151/dmgt.1378
|
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Nuttawoot Nupo, Chollawat Pookpienlert. Fractional domination and fractional total domination on Cayley digraphs of transformation semigroups with fixed sets[J]. AIMS Mathematics, 2024, 9(6): 14558-14573. doi: 10.3934/math.2024708
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