Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 4 2026 12 01 Independent domination, order, size, and maximum degree 1073 1080 15037 10.22049/cco.2025.30756.2609 EN Simone Dantas Instituto de Matemática e EstatÍstica, Universidade Federal Fluminense, Niterói, Brazil Michael A. Henning Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa Giovanna Arelis Baldeon Penao Instituto de Matemática e EstatÍstica, Universidade Federal Fluminense, Niterói, Brazil Dieter Rautenbach Institute of Optimization and Operations Research, Ulm University, Ulm, Germany Journal Article 2025 06 30 For a connected graph $G$ with $n>2\Delta$ vertices, $m$ edges, and maximum degree at most $\Delta\geq 3$, we show $i(G)\leq \left(1-\Omega\left(\frac{1}{\Delta^4}\right)\right)n-\frac{m}{\Delta}+O\left(\frac{1}{\Delta^2}\right)$ and discuss related problems.   https://comb-opt.azaruniv.ac.ir/article_15037_5228fbb71a2f533756a0da20b9e85b56.pdf