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Minimum Degree Orderings

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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.

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  1. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan

    Hiroshi Nagamochi

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  1. Hiroshi Nagamochi
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Correspondence to Hiroshi Nagamochi.

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Nagamochi, H. Minimum Degree Orderings. Algorithmica 56, 17–34 (2010). https://doi.org/10.1007/s00453-008-9239-2

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  • DOI: https://doi.org/10.1007/s00453-008-9239-2

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