Asieh Khoshnood; Dara Moazzami
Abstract
If we think of the graph as modeling a network, the vulnerability measurethe resistance of the network to disruption of operation after the failure of certainstations or communication links. In assessing the "vulnerability"of a graph one determines the extent to which the graph retains certainproperties ...
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If we think of the graph as modeling a network, the vulnerability measurethe resistance of the network to disruption of operation after the failure of certainstations or communication links. In assessing the "vulnerability"of a graph one determines the extent to which the graph retains certainproperties after the removal of vertices and / or edges. Many graph theoretical parameters have been used to describe the vulnerability of communication networks, including connectivity, integrity, toughness, binding number, tenacity and... . In this paper we survey and discuss tenacity and its properties in vulnerability calculation and we will comparedifferent measures of vulnerability with tenacity for several classes ofgraphs.
.Dara Moazzami
Abstract
The edge-tenacity $T_e(G)$ of a graph G was defined as\begin{center} $T_e(G)=\displaystyle \min_{F\subset E(G)}\{\frac{\mid F\mid +\tau(G-F)}{\omega(G-F)}\}$\end{center}where the minimum is taken over all edge cutset F of G. We defineG-F to be the graph induced by the edges of $E(G)-F$, $\tau(G-F)$is ...
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The edge-tenacity $T_e(G)$ of a graph G was defined as\begin{center} $T_e(G)=\displaystyle \min_{F\subset E(G)}\{\frac{\mid F\mid +\tau(G-F)}{\omega(G-F)}\}$\end{center}where the minimum is taken over all edge cutset F of G. We defineG-F to be the graph induced by the edges of $E(G)-F$, $\tau(G-F)$is the number of edges in the largest component of the graphinduced by G-F and $\omega(G-F)$ is the number of components of$G-F$. A set $F\subset E(G)$ is said to be a $T_e$-set of G if\begin{center} $T_e(G)=\frac{\mid F\mid+\tau(G-F)}{\omega(G-F)}$\end{center}Each component has at least one edge. In this paper we introducea new invariant edge-tenacity, for graphs. it is another vulnerability measure.we present several properties and bounds on the edge-tenacity. we alsocompute the edge-tenacity of some classes of graphs.
