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We say that $W$ is a $\\delta_k$-small set if $$ \\sqrt[k]{\\frac{\\sum_{v\\in W}d^k(v)}{\\abs W}}\\leq n-\\abs W. $$ Let $\\varphi^{(k)}(G)$ denote the smallest natural number $r$ such that $\\V(G)$ decomposes into $r$ $\\delta_k$-small sets, and let $\\alpha^{(k)}(G)$ denote the maximal number of vertices in a $\\delta_k$-small set of $G$. In this paper we obtain bounds for $\\alpha^{(k)}(G)$ and $\\varphi^{(k)}(G)$. 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