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Disproof of two conjectures on proper 2-dominating sets in graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 23، دوره 11، شماره 2، شهریور 2026، صفحه 713-715 اصل مقاله (251.13 K) | ||
| نوع مقاله: Short notes | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29967.2244 | ||
| نویسندگان | ||
| Mustapha Chellali* 1؛ Lutz Volkmann2 | ||
| 1LAMDA-RO Laboratory, Department of Mathematics, University of Blida, Blida, Algeria | ||
| 2Lehrstuhl II für Mathematik, RWTH Aachen, 52056 Aachen, Germany | ||
| چکیده | ||
| 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 $D$. | ||
| کلیدواژهها | ||
| 2-dominating sets؛ proper 2-dominating sets؛ proper 2-domination number | ||
| مراجع | ||
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[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. | ||
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