The problem of data transmission in communication network can betransformed into the problem of fractional factor existing in graph theory. Inrecent years, the data transmission problem in the specificnetwork conditions has received a great deal of attention, and itraises new demands to the corresponding mathematical model. Underthis background, many advanced results are presented on fractionalcritical deleted graphs and fractional ID deleted graphs. In thispaper, we determine that $G$ is a fractional\begin{document}$ (g,f,n',m) $\end{document}-critical deleted graph if\begin{document}$ δ(G)≥\frac{b^{2}(i-1)}{a}+n'+2m $\end{document}, \begin{document}$ n>\frac{(a+b)(i(a+b)+2m-2)+bn'}{a} $\end{document}, and$|N_{G}(x_{1})\cup N_{G}(x_{2})\cup···\cup N_{G}(x_{i})|≥\frac{b(n+n')}{a+b}$for any independent subset \begin{document}$ \{x_{1},x_{2},..., x_{i}\} $\end{document} of \begin{document}$ V(G) $\end{document}. Furthermore, the independent set neighborhood union condition for a graph to be fractional ID-\begin{document}$ (g,f,m) $\end{document}-deleted is raised. Some examples will be manifested to show the sharpness of independent set neighborhood union conditions.

In data transmission networks, the feasibility of data transmission can be characterized by the existence of fractional factors. If some channels and stations are not available in the transmission network at the moment, the possibility of transmission between data is characterized by whether the corresponding graph structure of the network is critical deleted. Toughness used to measure the vulnerability and robustness of a network, which is an important parameter to be considered in network designing. In this paper, we mainly study the relationship between toughness and the all fractional critical deleted graph, and a toughness condition for a graph to be all fractional (g,f,n)-critical deleted is determined.