Four Dominion Growth Regimes in Trees: Forcing, Fibonacci Enumeration, Periodicity, and Stability
Julian Allagan, Erin Gray, Jennifer Sawyer, Gabrielle Morgan
TL;DR
The paper investigates the dominion $\zeta(G)$, the number of minimum dominating sets, in trees to understand when optimal domination is rigid versus flexible. By analyzing pendant-path structures and complete binary trees, it identifies four distinct growth regimes, deriving exact formulas (e.g., $\gamma(G(n,r))=n$, $\zeta(G(n,r))=2^n$ for $r=1$) and a periodic, height-dependent law $\zeta(T_h)\in\{1,3\}$ for complete binary trees. It further establishes a sharp envelope for dominion under leaf deletions, $\zeta(T_h-X)\le 2^{m_1(X)}\zeta(T_h)$, with tightness demonstrated by single-leaf deletions. The work also provides linear-time dynamic programming techniques to compute $\gamma$ and $\zeta$ on trees and related path-based families, enabling efficient analysis of domination flexibility in large networks. Overall, the results offer a structural and algorithmic framework linking local forcing to global enumeration of optimal dominating sets, with applications in sensor placement, network resilience, and hierarchical monitoring.
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.