A Note on the Maximum Number of Minimal Connected Dominating Sets in a Graph

TL;DR

This paper establishes a new lower bound on the maximum number of minimal connected dominating sets in a graph, improving previous bounds and narrowing the gap in enumeration complexity.

Contribution

It provides a constructive proof that the maximum number of such sets grows at least as fast as 1.489^n, surpassing prior lower bounds.

Findings

Maximum number of minimal connected dominating sets is in (1.489^n)

Improves previous lower bound of (1.4422^n)

Reduces the gap between lower and upper bounds for enumeration

Abstract

We prove constructively that the maximum possible number of minimal connected dominating sets in a connected undirected graph of order $n$ is in $\Omega(1.489^n)$. This improves the previously known lower bound of $\Omega(1.4422^n)$ and reduces the gap between lower and upper bounds for input-sensitive enumeration of minimal connected dominating sets in general graphs as well as some special graph classes.