Minimal Dominating Sets in a Tree: Counting, Enumeration, and Extremal Results

TL;DR

This paper establishes an upper bound on the number of minimal dominating sets in a tree, introduces a semi-automatic method to compute this bound, and presents an efficient algorithm for listing all such sets.

Contribution

It provides the tightest known exponential bound on minimal dominating sets in trees and develops an output-sensitive enumeration algorithm based on dynamic programming.

Findings

Maximum of $95^{n/13}$ minimal dominating sets in an n-vertex tree

Growth constant approximately 1.4195, proven to be optimal

Linear-time algorithm for listing all minimal dominating sets

Abstract

A tree with $n$ vertices has at most $95^{n/13}$ minimal dominating sets. The growth constant $\lambda = \sqrt[13]{95} \approx 1.4194908$ is best possible. It is obtained in a semi-automatic way as a kind of "dominant eigenvalue" of a bilinear operation on sixtuples that is derived from the dynamic-programming recursion for computing the number of minimal dominating sets of a tree. We also derive an output-sensitive algorithm for listing all minimal dominating sets with linear set-up time and linear delay between successive solutions.