We study the dominion zeta(G), defined as the number of minimum dominating sets of a graph G, and analyze how local forcing and boundary effects control the flexibility of optimal domination in trees. For path-based pendant constructions, we identify a sharp forcing threshold: attaching a single pendant vertex to each path vertex yields complete independence with zeta = 2^gamma, whereas attaching two or more pendant vertices forces a unique minimum dominating set. Between these extremes, sparse pendant patterns produce intermediate behavior: removing endpoint pendants gives zeta = 2^(gamma - 2), while alternating pendant attachments induce Fibonacci growth zeta asymptotic to phi^gamma, where phi is the golden ratio. For complete binary trees T_h, we establish a rigid period-3 law zeta(T_h) in {1, 3} despite exponential growth in |V(T_h)|. We further prove a sharp stability bound under leaf deletions, zeta(T_h - X) <= 2^{m_1(X)} zeta(T_h), where m_1(X) counts parents that lose exactly one child; in particular, deleting a single leaf preserves the domination number and exactly doubles the dominion.

翻译：我们研究图G的最小支配集数量所定义的支配数zeta(G)，并分析局部强制与边界效应如何控制树中最优支配的灵活性。对于基于路径的悬挂构造，我们识别出一个尖锐的强制阈值：为路径的每个顶点连接单个悬挂顶点会产生完全独立性，此时zeta = 2^gamma；而连接两个或更多悬挂顶点则会强制产生唯一的最小支配集。在这两种极端情况之间，稀疏的悬挂模式会产生中间行为：移除端点悬挂顶点使得zeta = 2^(gamma - 2)，而交替的悬挂连接则引发斐波那契增长，即zeta渐近于phi^gamma，其中phi为黄金比例。对于完全二叉树T_h，我们建立了一个严格的周期-3规律：尽管|V(T_h)|呈指数增长，但zeta(T_h)的取值仅限于{1, 3}。我们进一步证明了在叶子删除下的尖锐稳定性界：zeta(T_h - X) <= 2^{m_1(X)} zeta(T_h)，其中m_1(X)统计恰好失去一个子节点的父节点数量；特别地，删除单个叶子节点会保持支配数不变，并使支配数恰好翻倍。