Toughness and fractional critical deleted graph (Q2811832)

A fractional \((g,f)\)-factor is a function \(h\) that assigns to each edge of a graph \(G\) a number in \([0,1]\) so that for each vertex \(x\), the total edge weight of \(G\) is bounded between \(g(x)\) and \(f(x)\), where \(g\) and \(f\) are two integer-valued functions defined on the vertices of \(G\). A graph \(G\) is said to be a fractional \((g,f)\)-deleted graph if for each edge \(e\), there exists a fractional \((g,f)\)-factor for \(G-e\). Finally, a graph \(G\) is called a fractional \((g,f,n)\)-critical graph if after deleting any \(n\) vertices from \(G\), the resulting graph still has a fractional \((g,f)\)-factor. The notion toughness introduced in the literature to measure the vulnerability of networks is used to determine a fractional \((g,f,n)\)-critical deleted graph. The paper also gives a toughness bound for fractional \((a,b,n)\)-critical deleted graphs.