Disproof of two conjectures on proper 2-dominating sets in graphs

Document Type : Short notes

Authors

1 LAMDA-RO Laboratory, Department of Mathematics, University of Blida, Blida, Algeria

2 Lehrstuhl II für Mathematik, RWTH Aachen, 52056 Aachen, Germany

Abstract

In this note, we disprove two conjectures recently stated on proper $2$-dominating sets in graphs. We recall that a proper $2$-dominating set of a graph $G=(V,E)$ is a subset $D$ of $V$ such that every vertex in $V-D$ has at least two neighbors in $D$ except for at least one vertex which must have exactly two neighbors in $D$.

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References

[1] P. Bednarz and M. Pirga, On proper 2-dominating sets in graphs, Symmetry 16 (2024), no. 3, Article ID: 296 https://doi.org/10.3390/sym16030296
[2] J.F. Fink and M.S. Jacobson, n-Domination in Graphs, Graph Theory with Applications to Algorithms and Computer Science (Y. Alavi and A.J. Schwenk, eds.), John Wiley & Sons, Inc., USA, 1985, pp. 283–300.
[3] A. Hansberg and L. Volkmann, Multiple Domination, Topics in Domination in Graphs (T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, eds.), Springer International Publishing, Cham, 2020, pp. 151–203.
Communications in Combinatorics and Optimization
Volume 11, Issue 2
June 2026
Pages 713-715

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