preprints.org > computer science and mathematics > computer vision and graphics > doi: 10.20944/preprints201607.0037.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 14 July 2016 / Approved: 15 July 2016 / Online: 15 July 2016 (09:50:33 CEST)

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 ${\gamma }_{2av}\left(G\right)=\frac{1}{\left|V\left(G\right)\right|}{\sum }_{v\in V\left(G\right)}{\gamma }_{2v}\left(G\right)$, 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.

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

Computer Science and Mathematics, Computer Vision and Graphics

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