# Domination number and minimum dominating sets in pseudofractal scale-free web and Sierpi\'nski graph

## TL;DR

This paper analytically determines the domination number and counts all minimum dominating sets in pseudofractal scale-free web and Sierpiński graph, revealing the influence of topology on the MDS problem.

## Contribution

It provides explicit solutions for the MDS problem in these specific graphs, highlighting the role of scale-free topology in dominating set properties.

## Key findings

- Pseudofractal scale-free web has a unique MDS.
- Domination number is half in the scale-free web compared to the Sierpiński graph.
- Scale-free topology influences the size and number of MDSs.

## Abstract

The minimum dominating set (MDS) problem is a fundamental subject of theoretical computer science, and has found vast applications in different areas, including sensor networks, protein interaction networks, and structural controllability. However, the determination of the size of a MDS and the number of all MDSs in a general network is NP-hard, and it thus makes sense to seek particular graphs for which the MDS problem can be solved analytically. In this paper, we study the MDS problem in the pseudofractal scale-free web and the Sierpi\'nski graph, which have the same number of vertices and edges. For both networks, we determine explicitly the domination number, as well as the number of distinct MDSs. We show that the pseudofractal scale-free web has a unique MDS, and its domination number is only half of that for the Sierpi\'nski graph, which has many MDSs. We argue that the scale-free topology is responsible for the difference of the size and number of MDSs between the two studied graphs, which in turn indicates that power-law degree distribution plays an important role in the MDS problem and its applications in scale-free networks.