Fractional arboricity, strength, and principal partitions in graphs and matroids

In a 1983 paper, D. Gusfield introduced a function which is called the strength of a graph or matroid. In terms of a graph G with edge set and at least one link, this is the function , where the minimum is taken over all subsets F of such that , the number of components of , is at least . In a 1986 paper, C. Payan introduced the fractional arboricity of a graph or matroid. In terms of a graph G with edge set and at least one link this function is , where H runs over