A Study on Total Rebellion Number in Graphs

V. Mohanaselvi, P. Shyamala Anto Mary, A. Monisha

Abstract


A set $R\subseteq V$ of a graph \(G = (V,E)\) is said to be a `rebellion set' (\(rb\)-set) of \(G\), if \(\arrowvert N_R(v) \arrowvert \leq \arrowvert N_{V/R}(v) \arrowvert\), for all \(v\in R\), \(\arrowvert R\arrowvert \geq \arrowvert V/R \arrowvert\) and \(\la R\ra\) has no isolated vertices. The total rebellion number \(\mathit{trb}(G)\) is the minimum cardinality of any total rebellion set in \(G\). A total rebellion set with cardinality \(\mathit{trb}(G)\) is denoted by \(\mathit{trb}(G)\)-set. In this paper, we defined the total rebellion number for simple graphs. Also, we determined its tight bounds for some standard graph and characterize this parameters.

Keywords


Rebellion number and Total rebellion number

Full Text:

PDF

References


F. Harary, Graph Theory, Addison Wesley, Reading (1969).

P. Kristiansen, S.K. Hedetniemi and S.T. Hedetniemi, Alliances in graphs, J. Cobin. Ath. Cobin. Copute. 48 (2004), 157 – 177.

T.W. Haynes, S.T. Hedetniemi and A. Henning, Global defensive alliance in graphs, Eletorn. J. Cobin. 10, Research paper 47, 2003.

V. Mohana Selvi and P. Shyamala Anto Mary, The rebellion number in graphs, International Journal of Scientific & Engineering Research 7(5) (May 2016), 56 – 62 (ISSN 2229-5518).

V. Mohana Selvi and P. Shyamala Anto Mary, National conference on bridging innovative trends in pure and applied mathematics, in: Independent Rebellion Number in Graphs, Bannari Amman Institute of Technology, Sathyamangalam (2016).

R. Kulli and B. Janakiram, The total domination number of a graph, Indian J. Pure Appl. Math. 27(6) (June 1996), 537 – 542.




DOI: http://dx.doi.org/10.26713%2Fjims.v9i3.938

eISSN 0975-5748; pISSN 0974-875X