EDGE DOMINATION NUMBER AND THE NUMBER OF MINIMUM EDGE DOMINATING SETS IN PSEUDOFRACTAL SCALE-FREE WEB AND SIERPIŃSKI GASKET

Xiaotian Zhou and Zhongzhi Zhang
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Xiaotian Zhou: Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai 200433, P. R. China2Shanghai Engineering Research Institute of Blockchain, School of Computer Science, Fudan University Shanghai 200433, P. R. China3Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, P. R. China
Zhongzhi Zhang: Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai 200433, P. R. China2Shanghai Engineering Research Institute of Blockchain, School of Computer Science, Fudan University Shanghai 200433, P. R. China3Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 07, 1-13

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 Sierpiński 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 Sierpiński gasket. Moreover, the number of all MEDSs in the pseudofractal scale-free web is also less than that corresponding to the Sierpiński gasket. We argue that the difference of the size and number of MEDSs between the two studied graphs lies in the scale-free topology.

Keywords: Minimum Edge Dominating Set; Edge Domination Number; Scale-Free Network; Sierpiński Gasket; Complex Network (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21502091

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