We consider random graphs with vertex degrees being drawn independently from a power-law distribution with parameter τ in the interval (1; 2). By computer simulation we study the resistance of such graphs to .target attacks. such as removal of vertices with the highest degree and to .random breakdowns., i.e. to equiprobable vertex removal. We show how graph structure characteristics (sizes of the giant and the second components, and the number of components) change with vertex deletion depending on the graph size N and parameter τ . The probability of graph destruction depending on N and τ was modeled for both cases.