Component Reliability Evaluation on Split-Stars

The failure of some key vertices in a network, due to either attacks or malfunctioning, may break the network into several disconnected <italic>components</italic>—the vertices within each component are connected, but there are no connections between components. When this happens, two measures should be of concern: (1) the number of components in the remaining network; and (2) the size of each component. When a given number of “break vertices” are removed from a network, we hope to have as few components as possible in the remaining network. On the other hand, it is certainly desirable that the components are as large as possible. Therefore both the number of components and the size of the maximal component after removing certain vertices can be a metric of a network’s <italic>fault tolerability</italic>, an important dimension of its robustness. In this paper, we examine the split-star, denoted <inline-formula> <tex-math notation="LaTeX">$S_{n}^{2}$ </tex-math></inline-formula>, in terms of this metric. We prove that when an arbitrary subset of vertices <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$|D|\leq 6n-15$ </tex-math></inline-formula> is removed from <inline-formula> <tex-math notation="LaTeX">$S_{n}^{2}$ </tex-math></inline-formula>, the remaining network will have at most 3 components, with the largest component having at least <inline-formula> <tex-math notation="LaTeX">$n!-|D|-3$ </tex-math></inline-formula> vertices. With <inline-formula> <tex-math notation="LaTeX">$|D|\leq 8n-21$ </tex-math></inline-formula>, the remaining network has at most 4 components, with the largest component’s size at least <inline-formula> <tex-math notation="LaTeX">$n!-|D|-5$ </tex-math></inline-formula>. As a theoretical application, the <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula><italic>-component connectivity</italic> of <inline-formula> <tex-math notation="LaTeX">$S_{n}^{2}$ </tex-math></inline-formula> is estimated by using the obtained maximal component and the minimal neighbor-set of independent set of size <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>. Moreover, we present a bound of <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula><italic>-component edge connectivity</italic> on <inline-formula> <tex-math notation="LaTeX">$S_{n}^{2}$ </tex-math></inline-formula> by considering the minimal neighbor edge-set of appropriate substructures in <inline-formula> <tex-math notation="LaTeX">$S_{n}^{2}$ </tex-math></inline-formula>.

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