All fractional (g,f)-factors in graphs

Let $G$ be a graph, and $g,f:V(G)\to N$ be two functions with $g(x)\le f(x)$ for each vertex $x$ in $G$. We say that $G$ has all fractional $(g,f)$-factors if $G$ includes a fractional $r$-factor for every $r:V(G)\to N$ such that $g(x)\le r(x)\le f(x)$ for each vertex $x$ in $G$. Let $H$ be a subgraph of $G$. We say that $G$ admits all fractional $(g,f)$-factors including $H$ if for every $r:V(G)\to N$ with $g(x)\leq r(x)\leq f(x)$foreachvertex$x$in$G$,$G$includesafractional$r$-factor$F_h$with$h(e)=1$forany$e\in E(H)$,thenwesaythat$G$ admits all fractional $(g,f)$-factors including $H$, where $h:E(G)\to [0,1]$ is the indicator function of $F_h$. In this paper, we obtain a characterization for the existence of all fractional $(g,f)$-factors including $H$ and pose a sufficient condition for a graph to have all fractional $(g,f)$-factors including $H$.