Independence number and the number of maximum independent sets in pseudofractal scale-free web and Sierpi\'nski gasket

TL;DR

This paper precisely calculates the independence number and counts all maximum independent sets in the pseudofractal scale-free web and Sierpiński gasket, revealing significant differences in their MIS properties.

Contribution

It provides exact solutions for the MIS problem in two specific graphs, highlighting their structural differences in maximum independent sets.

Findings

Pseudofractal scale-free web has twice the independence number of Sierpiński gasket.

The pseudofractal web has a unique maximum independent set.

Number of MISs in Sierpiński gasket grows exponentially with vertices.

Abstract

As a fundamental subject of theoretical computer science, the maximum independent set (MIS) problem not only is of purely theoretical interest, but also has found wide applications in various fields. However, for a general graph determining the size of a MIS is NP-hard, and exact computation of the number of all MISs is even more difficult. It is thus of significant interest to seek special graphs for which the MIS problem can be exactly solved. In this paper, we address the MIS problem in the pseudofractal scale-free web and the Sierpi\'nski gasket, which have the same number of vertices and edges. For both graphs, we determine exactly the independence number and the number of all possible MISs. The independence number of the pseudofractal scale-free web is as twice as the one of the Sierpi\'nski gasket. Moreover, the pseudofractal scale-free web has a unique MIS, while the number of MISs in the Sierpi\'nski gasket grows exponentially with the number of vertices.