Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

TL;DR

This paper analytically investigates maximum matchings and minimum dominating sets in Apollonian networks and their dual extended Tower of Hanoi graphs, revealing key combinatorial properties relevant to complex network applications.

Contribution

It provides exact formulas for matching and domination numbers, as well as counts of maximum matchings and minimum dominating sets in these networks, which were previously unknown.

Findings

Determined the matching number for both networks.

Calculated the domination number for both networks.

Counted the total maximum matchings and minimum dominating sets.

Abstract

The Apollonian networks display the remarkable power-law and small-world properties as observed in most realistic networked systems. Their dual graphs are extended Tower of Hanoi graphs, which are obtained from the Tower of Hanoi graphs by adding a special vertex linked to all its three extreme vertices. In this paper, we study analytically maximum matchings and minimum dominating sets in Apollonian networks and their dual graph- s, both of which have found vast applications in various fields, e.g. structural controllability of complex networks. For both networks, we determine their matching number, domination number, the number of maximum matchings, as well as the number of minimum dominating sets.