Optimal attack and reinforcement of a network
Journal of the ACM (JACM)
Computing the strength of a graph
SIAM Journal on Computing
Convex hulls of dense balanced graphs
Journal of Computational and Applied Mathematics - Special issue on asymptotic methods in analysis and combinatorics
Fractional arboricity, strength, and principal partitions in graphs and matroids
Discrete Applied Mathematics - Special issue: graphs in electrical engineering, discrete algorithms and complexity
A faster algorithm for computing the strength of a network
Information Processing Letters
Characterization of removable elements with respect to having k disjoint bases in a matroid
Discrete Applied Mathematics
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Let G be a non-trivial, loopless graph and for each non-trivial subgraph H of G, let g(H)=|E(H)||V(H)|-@w(H). The graph G is 1-balanced if @c(G), the maximum among g(H), taken over all non-trivial subgraphs H of G, is attained when H=G. This quantity @c(G) is called the fractional arboricity of the graph G. The value @c(G) appears in a paper by Picard and Queyranne and has been studied extensively by Catlin, Grossman, Hobbs and Lai. The quantity @c(G)-g(G) measures how much a given graph G differs from being 1-balanced. In this paper, we describe a systematic method of modifying a given graph to obtain a 1-balanced graph on the same number of vertices and edges. We obtain this by a sequence of iterations; each iteration re-defining one end-vertex of an edge in the given graph. After each iteration, either the value @c of the new graph formed is less than that of the graph from the previous iteration or the size of the maximal @c-achieving subgraph of the new graph is smaller than that of the graph in the previous iteration. We show that our algorithm is polynomial in time complexity. Further ways to decrease the number of iterations are also discussed.