Balanced Domination in Convex Polytopes, Trees, and Grid Graphs

TL;DR

This work studies the balanced domination number $\gamma_{bd}(G)$ via a linear-algebraic lens, treating balanced dominating functions (BDFs) as vectors in the kernel of $M(G)=A(G)+I(G)$ and noting that $\gamma_{bd}(G)=\max_{x\in\ker(M(G))} \sum_i x_i$. It proves a broad d-balanced outcome for graphs arising from convex polytopes $A_n$, $D_n$, and $R_n''$, and establishes grid graphs $Grid_{m\times n}$ with $3\le m\le n$ as d-balanced, providing the exact value $\gamma_{bd}(Grid_{m\times n})=0$. The paper also delivers structural results for rooted trees with two levels and for caterpillars, including necessary conditions for nonzero MBDFs and a parity-based leaf-count criterion $L(C_n)\equiv (3n-2) \pmod{4}$ when a nonzero MBDF exists. Together, these results resolve open questions about d-balancedness in several natural graph classes and outline a layer-based method to extend d-balancedness proofs to other families.

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.

Figures (4)

• Figure 1: Convex polytope $A_n$
• Figure 2: The graph of convex polytope $D_n$.
• Figure 3: The graph of convex polytope $R_n"$.
• Figure 4: Caterpillar graph with a spine of five vertices and 2, 3, 0, 2, and 4 leaves attached, respectively

Theorems & Definitions (23)

• Definition 1: xu2021balanced
• Definition 2: xu2021balanced
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof