Minimum degree orderings

  • Authors:

  • Hiroshi Nagamochi

  • Affiliations:

  • Department of Applied Mathematics and Physics, Kyoto University

  • Venue:
  • ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
  • Year:
  • 2007

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Abstract

It is known that, given an edge-weighted graph, a maximum adjacency ordering (MA ordering) of vertices can find a special pair of vertices, called a pendent pair, and that a minimum cut in a graph can be found by repeatedly contracting a pendent pair, yielding one of the fastest and simplest minimum cut algorithms. In this paper, we provide another ordering of vertices, called a minimum degree ordering (MD ordering) as a new fundamental tool to analyze the structure of graphs. We prove that an MD ordering finds a different type of special pair of vertices, called a flat pair, which actually can be obtained as the last two vertices after repeatedly removing a vertex with the minimum degree. By contracting flat pairs, we can find not only a minimum cut but also all extreme subsets of a given graph. These results can be extended to the problem of finding extreme subsets in symmetric submodular set functions.