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The criterion for invulnerability of a network based on the connectivity of a graph is well treated in literature. We define a criterion for survivability of a network in terms of the independence number of a graph. The following problems are then considered. 1) Find an -connected graph, with vertices and edges, whose independence number is the minimum possible, where denotes the greatest integer less than or equal to ) Given positive integers , and , find the realizability conditions for a graph with vertices and edges, whose connectivity is at least and independence number is at most .