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Graphs, Networks and Algorithms pp 405–439Cite as

Matchings

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Part of the book series: Algorithms and Computation in Mathematics ((AACIM,volume 5))

Abstract

This chapter deals with the problem of finding maximal matchings in arbitrary graphs; for the bipartite case, an efficient algorithms was already given in Sect. 7.2. In contrast to this case, it is not at all easy to reduce the general case to a ow problem, though this is possible (but beyond the scope of the present book). Nevertheless, we will see that the notion of an augmenting path can be modiffied appropriately to help enlarge a given matching. We will present the best known efficient algorithm, which rests on this concept and is due to Edmonds; this turns out to be much more difficult than in the bipartite case. We shall also present the most important theoretical results on matchings in general graphs: the 1-factor theorem of Tutte characterizing the graphs with a prefect matching, the more general Berge-Tutte formula giving the size of a maximal matching, and the Gallai-Edmonds structure theory.

And many a lovely flower and tree Stretch’d their blossoms out to me.

William Blake

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Notes

  1. 1.

    Kocay and Stone [KocSt93, KocSt95] showed that matchings may indeed be treated in the context of flow theory by introducing special types of networks and flows which satisfy certain symmetry conditions: balanced networks and balanced flows; related ideas can be found in the pioneering work of Tutte [Tut67] and in [GolKa96]. Subsequently, Fremuth-Paeger and Jungnickel provided a general theory based on this approach, including efficient algorithms; see [FreJu99a, FreJu99b, FreJu99c, FreJu01a, FreJu01b, FreJu01c, FreJu02, FreJu03]. We will not present this rather involved theory because that would take up far too much space; instead, we refer the reader to the original papers.

  2. 2.

    The edges of M′ are drawn bold in Fig. 13.4; note that \(M' \cap P'_{1} = M \cap P'_{1}\) and M′∩P 1⊂M⊕P 1.

  3. 3.

    Some authors use the terminology even vertex or odd vertex instead.

  4. 4.

    It is tempting to proceed by using all vertices of a blossom both as inner and as outer vertices, so that we cannot miss an augmenting path which uses part of a blossom. Indeed, Pape and Conradt [PapCo80] proposed splitting up each blossom into two alternating paths, so that the vertices of a blossom appear twice in the alternating tree T, both as inner and as outer vertices. Unfortunately, a serious problem arises: it might happen that an edge xy which was left out earlier in accordance with Case 3 (that is, an edge closing a cycle of even length) is needed at a later point because it is also contained in a blossom. The graph shown in Fig. 13.9 contains a unique augmenting path (between r and r′) which is not detected by the algorithm of [PapCo80]; thus their algorithm is too simple to be correct! This counterexample is due to Christian Fremuth-Paeger; see the first edition of the present book for more details.

  5. 5.

    Note that G/B is the result of a sequence of elementary contractions with respect to the edges contained in the blossom B; see Sect. 1.5.

  6. 6.

    In contrast to an ordinary BFS, we need the explicit distance function d(x) because contractions of blossoms may change the distances in the current tree T: a new pseudovertex b is, in general, closer to the root than some other active vertices which were earlier added to T; compare Example 13.4.1.

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  1. Institut für Mathematik, Universität Augsburg, Augsburg, Germany

    Dieter Jungnickel

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Jungnickel, D. (2013). Matchings. In: Graphs, Networks and Algorithms. Algorithms and Computation in Mathematics, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32278-5_13

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