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Amjadi, J., Ebadi, M., Sheikholeslami, Seyed Mahmoud, Volkmann, L.. (1405). Upper bounds for $[1,2]$-domination number in trees. , 11(4), 1163-1174. doi: 10.22049/cco.2026.31357.2831
J. Amjadi; M. Ebadi; Seyed Mahmoud Sheikholeslami; L. Volkmann. "Upper bounds for $[1,2]$-domination number in trees". , 11, 4, 1405, 1163-1174. doi: 10.22049/cco.2026.31357.2831
Amjadi, J., Ebadi, M., Sheikholeslami, Seyed Mahmoud, Volkmann, L.. (1405). 'Upper bounds for $[1,2]$-domination number in trees', , 11(4), pp. 1163-1174. doi: 10.22049/cco.2026.31357.2831
Amjadi, J., Ebadi, M., Sheikholeslami, Seyed Mahmoud, Volkmann, L.. Upper bounds for $[1,2]$-domination number in trees. , 1405; 11(4): 1163-1174. doi: 10.22049/cco.2026.31357.2831

Upper bounds for $[1,2]$-domination number in trees

Communications in Combinatorics and Optimization
مقاله 9، دوره 11، شماره 4، اسفند 2026، صفحه 1163-1174    اصل مقاله (369.8 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2026.31357.2831
نویسندگان
J. Amjadi* 1؛ M. Ebadi1؛ Seyed Mahmoud Sheikholeslami* 1؛ L. Volkmann2
1Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran
2Institute for Geometry and Practical Mathematics, RWTH Aachen University, 52056 Aachen, Germany
چکیده
A set $S$ of vertices is a $[1,2]$-set of a graph $G$ if every vertex $v$ not in $S$ is adjacent to at least one but no more than two vertices in $S$. The minimum cardinality of a $[1,2]$-set is the $[1,2]$-domination number. In this paper, we present two upper bounds on the $[1,2]$-domination number of trees in terms of the order, number of support vertices and number of leaves. Furthermore, extremal trees reaching one of these two bounds are provided.
کلیدواژه‌ها
$[1,2]$-set؛ $[1,2]$-domination number؛ trees
مراجع
[1] J. Cáceres, C. Hernando, M. Mora, I.M. Pelayo, and M.L. Puertas González, Perfect and quasiperfect domination in trees, Appl. Analysis Discrete Math. 10 (2016), 46–64. https://doi.org/10.2298/AADM160406007C

[2] Y. Caro, A. Hansberg, and M. Henning, Fair domination in graphs, Discrete Math. 312 (2012), 2905–2914. https://doi.org/10.1016/j.disc.2012.05.006

[3] M. Chellali, T.W. Haynes, S.T. Hedetniemi, and A. McRae, $[1, 2]$-sets in graphs, Discrete Appl. Math. 161 (2013), 2885–2893.  https://doi.org/10.1016/j.dam.2013.06.012

[4] M. Livingston and Q.F. Stout, Perfect dominating sets, Congr. Numer. 79 (1990), 187–203.

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