树中的四种支配增长模式：强制、斐波那契计数、周期性和稳定性
Four Dominion Growth Regimes in Trees: Forcing, Fibonacci Enumeration, Periodicity, and Stability
AI总结：
本文研究了树中支配集的增长模式，揭示了不同构造方式对支配集数量的影响，并证明了完全二叉树中支配域的周期性规律及稳定性边界。
摘要：
我们研究了支配ζ函数，定义为图G的最小支配集数量，并分析局部强制和边界效应如何控制树中最优支配的灵活性。对于基于路径的悬垂构造，我们确定了一个尖锐的强制阈值：在每个路径顶点上附加一个悬垂顶点会产生完全独立性，ζ=2^γ；而附加两个或更多悬垂顶点会强制产生唯一的最小支配集。在这些极端之间，稀疏悬垂模式产生中间行为：移除端点悬垂会得到ζ=2^(γ-2)，而交替悬垂附件会诱导斐波那契增长，ζ渐近趋于φ^γ，其中φ是黄金比例。对于完全二叉树T_h，我们建立了一个刚性的周期-3定律，ζ(T_h)在{1,3}之间，尽管|V(T_h)|呈指数增长。我们进一步证明了在叶删除下的尖锐稳定性界限，ζ(T_h - X) ≤ 2^{m_1(X)} ζ(T_h)，其中m_1(X)计数失去恰好一个子节点的父节点；特别是，删除一个叶节点会保持支配数并恰好翻倍支配域。
英文摘要：
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.
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