Conservative weightings and ear-decompositions of graphs

Abstract

A subsetJ of edges of a connected undirected graphG=(V, E) is called ajoin if |CJ|≤|C|/2 for every circuitC ofG. Answering a question of P. Solé and Th. Zaslavsky, we derive a min-max formula for the maximum cardinality μ of a joint ofG. Namely, μ=(φ+|V|−1)/2 where φ denotes the minimum number of edges whose contraction leaves a factor-critical graph.

To study these parameters we introduce a new decomposition ofG, interesting for its own sake, whose building blocks are factor-critical graphs and matching-covered bipartite graphs. We prove that the length of such a decomposition is always φ and show how an optimal join can be constructed as the union of perfect matchings in the building blocks. The proof relies on the Gallai-Edmonds structure theorem and gives rise to a polynomial time algorithm to construct the optima in question.

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Frank, A. Conservative weightings and ear-decompositions of graphs. Combinatorica 13, 65–81 (1993). https://doi.org/10.1007/BF01202790

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AMS subject classification code (1991)

  • 05 C 70
  • 05 C 75
  • 94 B 60