# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.04517