Edge Domination Number and the Number of Minimum Edge Dominating Sets in Pseudofractal Scale-Free Web and Sierpi\'nski Gasket

TL;DR

This paper analytically investigates the minimum edge dominating set problem in pseudofractal scale-free web and Sierpiński gasket, deriving exact formulas for edge domination number and the count of MEDSs, highlighting topological impacts.

Contribution

It provides exact solutions for the MEDS problem in two specific complex graphs, revealing how topology influences MEDS size and quantity.

Findings

Edge domination number is one-ninth of total edges in pseudofractal web.

Number of MEDSs is fewer in pseudofractal web than in Sierpiński gasket.

Topology differences explain variations in MEDS properties.

Abstract

As a fundamental research object, the minimum edge dominating set (MEDS) problem is of both theoretical and practical interest. However, determining the size of a MEDS and the number of all MEDSs in a general graph is NP-hard, and it thus makes sense to find special graphs for which the MEDS problem can be exactly solved. In this paper, we study analytically the MEDS problem in the pseudofractal scale-free web and the Sierpi\'nski gasket with the same number of vertices and edges. For both graphs, we obtain exact expressions for the edge domination number, as well as recursive solutions to the number of distinct MEDSs. In the pseudofractal scale-free web, the edge domination number is one-ninth of the number of edges, which is three-fifths of the edge domination number of the Sierpi\'nski gasket. Moreover, the number of all MEDSs in the pseudofractal scale-free web is also less than that corresponding to the Sierpi\'nski gasket. We argue that the difference of the size and number of MEDSs between the two studied graphs lies in the scale-free topology.