Vertex Removal on Perfect Italian Domination and $\gamma_I^p$-Stability of Graphs

Abstract

Perfect Italian Domination is a domination concept where all vertices are assigned one of the labels among $0$, $1$ and $2$ such that the sum of the labels in the neighbourhood of every vertex labelled $0$ should be exactly $2$. If the zero-labelled vertices are adjacent to any other vertex, they should all be zero-labelled. We examine a few graph classes and discuss in detail the criticality concept of Perfect Italian Domination. We also define $\gamma_I^p-$ stable graphs and PID critical graphs. Following our definitions of $\gamma_I^p$-stable and PID critical graphs, we have grouped some graph classes. We characterise a family of trees that is $\gamma_I^p$-stable.

Keywords

• Leaf vertex
• Perfect Italian Domination
• Stem vertex

References

1. Banerjee, S., Henning, M.A., Pradhan, D., Perfect Italian domination in cographs, Applied Mathematics and Computation 391(2021).
2. Haynes, T.W., Hedetniemi, S., Slater, P., Fundamentals of Domination in Graphs, CRC Press, 1998.
3. Haynes, T.W., Henning, M.A. Perfect Italian domination in trees, Discrete Applied Mathematics 260(2019), 164–177.
4. Haynes, T.W., Hedetniemi, S.T., Henning, M.A., Structures of Domination in Graphs, 2021.
5. Henning, M.A., Klostermeyer, W.F., Italian domination in trees, Discrete Applied Mathematics, 217(2017), 557–564.
6. Lauri, J., Mitillos, C., Perfect Italian domination on planar and regular graphs, Discrete Applied Mathematics 285(2020), 676–687.
7. Mojdeh, D.A., Parsian, A., Masoumi, I., Strong Roman domination number of complementary prism graphs, Turkish Journal of Mathematics and Computer Science 11(1)(2019), 40–47.
8. Poovathingal, A., Kureethara, J. V., On some graphs whose domination number is the Perfect Italian domination number, Congress on Intelligent Systems, (2023), 191–200.

9. Poovathingal, A., Kureethara J.V., Modelling networks with attached storage using Perfect Italian domination, International Conference on MAchine inTelligence for Research & Innovations, (2023), 23–33.
10. Pradhan, D., Banerjee, S., Liu, J.B., Perfect Italian domination in graphs: Complexity and algorithms, Discrete Applied Mathematics 319(2022), 271–295.
11. Sumner, D.P., Critical concepts in domination, Annals of Discrete Mathematics, 48(1991), 33–46.
12. Varghese, J., Aparna Lakshmanan, S., Perfect Italian domination number of graphs, arXiv preprint arXiv:1910.12260, (2019).
13. Varghese, J., Anu, V., Aparna Lakshmanan, S., Italian domination and perfect Italian domination on Sierpinski graphs, Journal of Discrete Mathematical Sciences and Cryptography, 24(2021), 1885–1894.