Balanced Domination in Convex Polytopes, Trees, and Grid Graphs

TL;DR

This paper investigates the balanced domination number across various graph classes, establishing new results for convex polytopes, trees, and grid graphs, and providing characterizations and exact values.

Contribution

It introduces the concept of d-balancedness for new graph classes and characterizes it for rooted trees and binary trees, also determining the exact domination number for grid graphs.

Findings

Convex polytopes A_n, D_n, Rn'' are d-balanced.

Full binary trees are d-balanced.

Exact balanced domination number for 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 A_n, D_n, 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.