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Spanning Trees

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Graphs, Networks and Algorithms

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

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Abstract

We have already encountered the notion of a tree in Chap. 1 and obtained some basic results, including the number of trees on n vertices. In the present chapter, we will study this important class of graphs in considerably more detail. After some further characterizations of trees, we shall study another way of determining the number of trees on n vertices which actually applies more generally: it allows one to compute the number of spanning trees in any given connected graph. Following these theoretical results, the major part of the chapter will be devoted to a network optimization problem, namely to finding a spanning tree for which the sum of all edge lengths is minimal. This problem has many applications; for example, the vertices might represent cities we want to connect to a system supplying electricity; then the edges represent the possible connections and the length of an edge states how much it would cost to build that connection. Other possible interpretations are tasks like establishing traffic connections (for cars, trains or planes) or designing a network for TV broadcasts. We shall present an interesting characterization of minimal spanning trees and use this criterion to establish the most important algorithms for determining such a tree, namely those of Prim, Kruskal, and Boruvka. Following this, we discuss several further applications (e.g., the bottleneck problem) and spanning trees with additional restrictions. We will also consider Steiner trees (these are trees where it is allowed to add some new vertices) and arborescences (directed trees).

I think that I shall never see

A poem lovely as a tree.

Joyce Kilmer

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Notes

  1. 1.

    We will treat this result in Chap. 6. Actually we shall use a different proof which is not based on Theorem 4.2.5.

  2. 2.

    Beware: some authors use the term Steiner tree for what we call a minimal Steiner tree. As an exercise, the reader might try to settle the geometric Steiner tree problem for the vertices of a unit square: here one gets two Steiner points, and the minimal Steiner tree has length \(1+\sqrt{3}\). See [Cox61], Sect. 1.8, or [CouRo41], p. 392.

  3. 3.

    Here is an exercise for those who remember their high school geometry. Prove that the Fermat point of a triangle in which no angle exceeds 120∘ is the unique point from which the three sides each subtend a 120∘ angle. See, for example, [Cox61], Sect. 1.8.

  4. 4.

    As some of the arithmetic operations concerned are exponentiations, this estimate of the complexity might be considered a little optimistic.

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

    Dieter Jungnickel

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Jungnickel, D. (2013). Spanning Trees. 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_4

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