Optimal attack and reinforcement of a network
Journal of the ACM (JACM)
Equilibrium graphs and rational trees
European Journal of Combinatorics
Fractional arboricity, strength, and principal partitions in graphs and matroids
Discrete Applied Mathematics - Special issue: graphs in electrical engineering, discrete algorithms and complexity
On the higher-order edge toughness of a graph
Discrete Mathematics
On the spanning tree packing number of a graph: a survey
Discrete Mathematics
Transforming a graph into a 1-balanced graph
Discrete Applied Mathematics
Graph Theory
Balanced and 1-balanced graph constructions
Discrete Applied Mathematics
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The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with k edge-disjoint spanning trees. Edmonds generalizes this theorem to matroids with k disjoint bases. For any graph G that may not have k-edge-disjoint spanning trees, the problem of determining what edges should be added to G so that the resulting graph has k edge-disjoint spanning trees has been studied by Haas (2002) [11] and Liu et al. (2009) [17], among others. This paper aims to determine, for a matroid M that has k disjoint bases, the set E"k(M) of elements in M such that for any e@?E"k(M), M-e also has k disjoint bases. Using the matroid strength defined by Catlin et al. (1992) [4], we present a characterization of E"k(M) in terms of the strength of M. Consequently, this yields a characterization of edge sets E"k(G) in a graph G with at least k edge-disjoint spanning trees such that @?e@?E"k(G), G-e also has k edge-disjoint spanning trees. Polynomial algorithms are also discussed for identifying the set E"k(M) in a matroid M, or the edge subset E"k(G) for a connected graph G.