The Tutte's condition in terms of graph factors

Let $G$ be a connected general graph of even order, with a function $f\colon V(G)\to\Z^+$\mathrm{.}Weobtainthat$G$ satisfies the Tutte’s condition $$o(G-S)\le \sum _{v\in S}f(v)\text{for any nonempty set }S\subset V(G),$$ with respect to $f$ if and only if $G$ contains an $H$-factor for any function $H\text{\hspace{0.17em}⁣}:V(G)\to 2^{\mathbb{N}}$ such that $H(v)\in \{J_f(v),J_f^{+}(v)\}$ for each $v\in V(G)$,wheretheset$J_f(v)$consistsoftheinteger$f(v)$ and all positive odd integers less than $f(v)$, and the set $J_f^{+}(v)$ consists of positive odd integers less than or equal to $f(v)+1$. We also obtain a characterization for graphs of odd order satisfying the Tutte’s condition with respect to a function.