Bounds on the Maximum Number of Minimum Dominating Sets

TL;DR

This paper establishes probabilistic lower bounds on the maximum number of minimum dominating sets in graphs with a given domination number, using random graph models and matrix methods.

Contribution

It introduces new probabilistic and matrix-based techniques to bound the number of minimum dominating sets in graphs with specified domination numbers.

Findings

Almost all sets of size 3 dominate the graph w.h.p.

Provides lower bounds on the number of non-dominating sets using a modified adjacency matrix.

Demonstrates the effectiveness of probabilistic methods in combinatorial graph bounds.

Abstract

We use probabilistic methods to find lower bounds on the maximum number, in a graph with domination number \gamma, of dominating sets of size \gamma. We find that we can randomly generate a graph that, w.h.p., is dominated by almost all sets of size \gamma. At the same time, we use a modified adjacency matrix to obtain lower bounds on the number of sets of a given size that do not dominate a graph on n vertices