The average lower reinforcement number of a graph

Turaci, Tufan; Aslan, Ersin

doi:10.1051/ita/2016015

Turaci, Tufan ¹ ; Aslan, Ersin ²

¹ Department of Mathematics, Faculty of Science, Karabük University, 78050, Karabük, Turkey.
² Turgutlu Vocational Training School, Celal Bayar University, 45400 Manisa, Turkey.

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 50 (2016) no. 2, pp. 135-144.

Résumé

Let $G = (V (G), E (G))$ be a simple undirected graph. The reinforcement number of a graph is a vulnerability parameter of a graph. We have investigated a refinement that involves the average lower reinforcement number of this parameter. $𝑙𝑜𝑤𝑒𝑟 𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑚𝑒𝑛𝑡 𝑛𝑢𝑚𝑏𝑒𝑟$ , denoted by $r_{e^{*}} (G)$ , is the minimum cardinality of $𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑚𝑒𝑛𝑡 𝑠𝑒𝑡$ in $G$ that contains the edge $e^{*}$ of the complement graph $\bar{G}$ . The $𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑙𝑜𝑤𝑒𝑟 𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑚𝑒𝑛𝑡 𝑛𝑢𝑚𝑏𝑒𝑟$ of $G$ is defined by $r_{a v} (G) = \frac{1}{| E (\bar{G}) |} Σ_{e^{*} \in E (\bar{G})} r_{e^{*}} (G)$ . In this paper, we define the average lower reinforcement number of a graph and we present the exact values for some well−known graph families.

MR Zbl

DOI : 10.1051/ita/2016015

Classification : 05C40, 05C69, 68M10, 68R10
Mots clés : Graph vulnerability, connectivity, network design and communication, domination number, reinforcement number, average lower reinforcement number

Affiliations des auteurs :

Turaci, Tufan ¹ ; Aslan, Ersin ²

¹ Department of Mathematics, Faculty of Science, Karabük University, 78050, Karabük, Turkey.
² Turgutlu Vocational Training School, Celal Bayar University, 45400 Manisa, Turkey.

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     title = {The average lower reinforcement number of a graph},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {135--144},
     publisher = {EDP-Sciences},
     volume = {50},
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Turaci, Tufan; Aslan, Ersin. The average lower reinforcement number of a graph. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 50 (2016) no. 2, pp. 135-144. doi : 10.1051/ita/2016015. http://archive.numdam.org/articles/10.1051/ita/2016015/

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