Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 4 2026 12 01 Upper bounds for $[1,2]$-domination number in trees 1163 1174 15128 10.22049/cco.2026.31357.2831 EN J. Amjadi Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran M. Ebadi Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran Seyed Mahmoud Sheikholeslami Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran L. Volkmann Institute for Geometry and Practical Mathematics, RWTH Aachen University, 52056 Aachen, Germany Journal Article 2025 12 23 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. https://comb-opt.azaruniv.ac.ir/article_15128_2250a82aa247078a96542171dcafe737.pdf