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Graphs, Networks and Algorithms pp 251–273Cite as

Connectivity and Depth First Search

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

Abstract

In previous chapters, we encountered the notions of connectivity, strong connectivity, and k-connectivity. In particular, we already know an efficient method for determining the connected components of a graph: breadth first search (BFS). In the present chapter, we mainly treat algorithmic questions concerning k-connectivity and strong connectivity. To this end, we introduce a further important strategy for searching graphs and digraphs (besides BFS), namely depth first search—which may also be thought of as a strategy for traversing a maze. In addition, we present various theoretical results, such as characterizations of 2-connected graphs and of edge connectivity.

How beautiful the world would be if there were a rule for getting around in labyrinths.

Umberto Eco

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Notes

  1. 1.

    For the time being, we leave it to the reader to prove this claim; alternatively, see Theorem 8.3.1.

  2. 2.

    Recall that a stack is a list where elements are appended at the end and removed at the end as well (last in—first out), in contrast to a queue where elements are appended at the end, but removed at the beginning (first in—first out). For a more detailed discussion of these data structures (as well as for possible implementations), we refer to [AhoHU74, AhoHU83] or to [CorLRS09].

  3. 3.

    This means replacing each edge uv of G by vu.

  4. 4.

    Some authors use the terms line connectivity and line connected instead.

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

    Dieter Jungnickel

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Jungnickel, D. (2013). Connectivity and Depth First Search. 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_8

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