Balanced Domination in Convex Polytopes, Trees, and Grid Graphs
Abstract
This paper addresses two open questions posed in [27] regarding the balanced domination number in graphs. We show that three new classes of graphs, those of convex polytopes An, Dn, and Rn'', 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. Several results for caterpillar graphs are established. Moreover, we determine and prove the exact balanced domination number for grid graphs. Finally, we conclude by providing several open problems of interest.
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