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

TL;DR

This paper explores the combinatorial properties of minimum dominating sets in trees, revealing growth regimes, periodic behaviors, and stability bounds under modifications, with implications for graph domination theory.

Contribution

It introduces new growth regimes and periodicity laws for dominion zeta in trees, and establishes stability bounds under leaf deletions, advancing understanding of domination dynamics.

Findings

Identified a sharp forcing threshold for pendant attachments affecting zeta.

Discovered Fibonacci growth in certain pendant configurations.

Proved a period-3 law for complete binary trees and stability bounds under leaf deletions.

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.