Abstract
A. Frank described in [1] an algorithm to determine the minimum number of edges in a graph G whose contraction leaves a factor-critical graph and he asked if there was an algorithm for the weighted version of the problem. We prove that the minimal critical-making edge-sets form the bases of a matroid and hence the matroid greedy algorithm gives rise to the desired algorithm.
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References
A. Frank: Conservative weightings and ear-decompositions of graphs,Combinatorica,13 (1) (1993), 65–81.
L. Lovász, andM. D. Plummer:Matching Theory, Akadémiai Kiadó, Budapest, 1986.
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Z. Szigeti: On ear-decompositions and critical-making sets, 1992, (in Hungarian).
Z. Szigeti: On Lovász's Cathedral Theorem,Proceedings of the Third Conference on Integer Programming and Combinatorial Optimization, eds.: G. Rinaldi and L. A. Wolsey, 1993, 413–423.
Z. Szigeti: Conservative weightings of graphs, Ph. D. thesis, 1994.
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Partially supported by OTKA F014919, OTKA T17181 and OTKA T17580.