A counterexample to a conjecture of Jafari Rad and Volkmann

Document Type : Short notes

Authors

Department of Mathematics, University of Blida 1. B.P. 270, Blida, Algeria

Abstract

In this short note, we disprove the conjecture of Jafari Rad and Volkmann
that every $\gamma $-vertex critical graph is $\gamma _{R}$-vertex critical,
where $\gamma (G)$ and $\gamma _{R}(G)$ stand for the domination number and
the Roman domination number of a graph $G$, respectively.

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References

[1] R.C. Brigham, P.Z. Chinn, and R.D. Dutton, Vertex domination-critical graphs, Networks 18 (1988), no. 3, 173–179.
[2] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Roman domination in graphs, Topics in Domination in Graphs (T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, eds.), Springer International Publishing, Berlin, 2020, pp. 365–
409.
[3] N. Jafari Rad, A. Hansberg, and L. Volkmann, Vertex and edge critical Roman domination in graphs, Util. Math. 92 (2013), 73–88.
[4] N. Jafari Rad and L. Volkmann, Changing and unchanging the Roman domination number of a graph, Util. Math. 89 (2012), 79–95.
Communications in Combinatorics and Optimization
Volume 7, Issue 1
June 2022
Pages 91-92

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