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Recognizing $\\mathbf{W_1}$ graphs is co-NP-hard, as shown by Chv\\'atal and Slater (1993) and, independently, by Sankaranarayana and Stewart (1992). Extending this result and answering a recent question of Levit and Tankus, we show that recognizing $\\mathbf{W_k}$ graphs is co-NP-hard for $k \\geq 2$. 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