Four Dominion Growth Regimes in Trees: Forcing, Fibonacci Enumeration, Periodicity, and Stability

Abstract: 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.