Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs

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$ (g,f,n',m) $-critical deleted graph ifδ(G)≥b²(i-1)a+n'+2m, n>(a+b)(i(a+b)+2m-2)+bn'/a, and {document}|NG(x₁)∪ NG(x₂)∪···∪ NG(xᵢ)|≥b(n+n')/a+b {document} for any independent subset ₁,x₂,..., xᵢ\ of $ V(G) $. Furthermore, the independent set neighborhood union condition for a graph to be fractional ID-$ (g,f,m) $-deleted is raised. Some examples will be manifested to show the sharpness of independent set neighborhood union conditions.