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Let $G$ be a graph with vertex set $V(G)$ and let $H:V(G)\rightarrow 2^N$ bea set function associating with $G$. An $H$-factor of graph $G$ is a spanningsubgraphs $F$ such that $$d_F(v)\in H(v){4em}\hbox{for every}v\in V(G).$$ Let$f:V(G)\rightarrow N$ be an even integer-valued function such that $f\geq 4$and let $H_f(v)=\{1,3,...,f(v)-1, f(v)\}$ for $v\in V(G)$. In this paper, weinvestigate $H_f$-factors of graphs $G$ by using Lov\'asz's structuraldescriptions. Let $o(G)$ denote the number of odd components of $G$.  We show that if one of the following conditions holds, then $G$ contains an$H_f$-factor. [$(i)$] $o(G-S)\leq f(S)$ for all $S\subseteq V(G)$; [$(ii)$]$|V(G)|$ is odd, $d_G(v)\geq f(v)-1$ for all $v\in V(G)$ and $o(G-S)\leq f(S)$for all $\emptyset\neq S\subseteq V(G)$. As a corollary, we show that if agraph $G$ with odd order and minimum degree $2n-1$ satisfies $$o(G-S)\leq2n|S|{4em}{for all}  \emptyset\neq S\subseteq V(G),$$ then $G$ contains an $H_n$-factor. Inparticular, we make progress on the characterization problem for a specialfamily of graphs proposed by Akiyama and Kano.

An Extension of Cui-Kano's Characterization Problem on Graph Factors