Let  where  is a signed graph. A set  is said to be a restrained dominating set of  if D is a restrained dominating set of  and every cycle formed by the edges across D to VD and within VD is balanced. The cardinality of a minimum restrained dominating set in Σ is the restrained domination number of Σ and is denoted by  In this paper, we investigate how the addition of an edge between any two non-adjacent vertices or the removal of a vertex from any signed graph affects the restrained domination number for heterogeneous signed graphs.

signed graphs, restrained domination, critical abundant, vertex removal.

Received: November 29, 2022; Accepted: January 24, 2023; Published: March 15, 2023

How to cite this article: A. J. Mathias, V. Sangeetha and M. Acharya, Restrained critical and abundant signed graphs, Advances and Applications in Discrete Mathematics 38(1) (2023), 49-68. http://dx.doi.org/10.17654/0974165823018

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2(2) (1953), 143-146.
[2] T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4(1) (1982), 47-74.
[3] B. D. Acharya, Domination and absorbance in signed graphs and digraphs: I. foundations, J. Combin. Math. Combin. Comput. 84 (2013), 5-20.
[4] A. J. Mathias, V. Sangeetha and M. Acharya, Restrained domination in signed graphs, Acta Univ. Sapientiae Math. 12(1) (2020), 155-163.
[5] T. W. Haynes, S. Hedetniemi and P. Slater, Fundamentals of Domination in Graphs, CRC Press, 2013.
[6] D. P. Sumner, Critical concepts in domination, Discrete Math. 86 (1990), 33-46.
[7] H. Walikar and B. Acharya, Domination critical graphs, National Academy Science Letters 2(2) (1979), 70-72.
[8] T. N. Vasantha, A study on restrained domination number of a graph, Ph. D. dissertation, Manonmaniam Sundaranar University, Tirunelveli, 2007.
[9] A. J. Mathias, V. Sangeetha and M. Acharya, Critical concepts of restrained domination in signed graphs, Discrete Math. Algorithms Appl. 14(6) (2022), Paper No. 2250010, 11 pp.
[10] B. Adhikari, A. Singh and S. K. Yadav, Corona product of signed graphs and its application to signed network modelling, 2019. arXiv preprint arXiv:1908.10018.