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 $D$.
[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.
Articles in Press, Accepted Manuscript Available Online from 10 September 2024
Chellali, M., & Volkmann, L. (2024). Disproof of two conjectures on proper 2-dominating sets in graphs. Communications in Combinatorics and Optimization, (), -. doi: 10.22049/cco.2024.29967.2244
MLA
Mustapha Chellali; Lutz Volkmann. "Disproof of two conjectures on proper 2-dominating sets in graphs", Communications in Combinatorics and Optimization, , , 2024, -. doi: 10.22049/cco.2024.29967.2244
HARVARD
Chellali, M., Volkmann, L. (2024). 'Disproof of two conjectures on proper 2-dominating sets in graphs', Communications in Combinatorics and Optimization, (), pp. -. doi: 10.22049/cco.2024.29967.2244
VANCOUVER
Chellali, M., Volkmann, L. Disproof of two conjectures on proper 2-dominating sets in graphs. Communications in Combinatorics and Optimization, 2024; (): -. doi: 10.22049/cco.2024.29967.2244