Balanced domination in convex polytopes, trees, and grid graphs

This paper addresses two open questions posed in Xu et al. (2021) regarding the balanced domination number in graphs. We show that three new classes of graphs—those of convex polytopes A n , D n , and R n ′ ′ —are d -balanced. Further, we provide a characterization of d -balancedness for rooted trees with two levels of descendants and prove that each full binary tree is d -balanced. Additionally, several results for caterpillar graphs are established. Moreover, we determine and prove the exact balanced domination number on grid graphs. Finally, we conclude the paper by providing several new problems of interest.