Preprint Article Version 1 NOT YET PEER-REVIEWED

THE AVERAGE LOWER 2-DOMINATION NUMBER OF WHEELS RELATED GRAPHS AND AN ALGORITHM

1. Department of Mathematics, Faculty of Science, Karabük University, Karabük 78050, Turkey
Version 1 : Received: 14 July 2016 / Approved: 15 July 2016 / Online: 15 July 2016 (09:50:33 CEST)

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: TURACI, T. THE AVERAGE LOWER 2-DOMINATION NUMBER OF WHEELS RELATED GRAPHS AND AN ALGORITHM. Preprints 2016, 2016070037 (doi: 10.20944/preprints201607.0037.v1). TURACI, T. THE AVERAGE LOWER 2-DOMINATION NUMBER OF WHEELS RELATED GRAPHS AND AN ALGORITHM. Preprints 2016, 2016070037 (doi: 10.20944/preprints201607.0037.v1).

A peer-reviewed article of this Preprint also exists.

Journal reference: Mathematical and Computational Applications 2016, 21, 29
DOI: 10.3390/mca21030029

Abstract

The problem of quantifying the vulnerability of graphs has received much attention nowadays, especially in the field of computer or communication networks. In a communication network, the vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modeling a network, the average lower 2-domination number of a graph is a measure of the graph vulnerability and it is defined by ${\gamma }_{2av}\left(G\right)=\frac{1}{|V\left(G\right)|}{\sum }_{v\in V\left(G\right)}{\gamma }_{2v}\left(G\right)$, where the lower 2-domination number, denoted by ${\gamma }_{2v}\left(G\right)$, of the graph G relative to v is the minimum cardinality of 2-domination set in G that contains the vertex v. In this paper, the average lower 2-domination number of wheels and some related networks namely gear graph, friendship graph, helm graph and sun flower graph are calculated. Then, we offer an algorithm for computing the 2-domination number and the average lower 2-domination number of any graph G.

Subject Areas

graph vulnerability; connectivity; network design and communication; domination number; average lower 2-domination number