Graphs, Networks and Algorithms by Dieter Jungnickel

Author:Dieter Jungnickel
Language: eng
Format: epub, pdf
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

Proof

By Theorem 2.6.6, we may check with complexity O(|E|) whether H is acyclic; in this case, μ(H,w)=∞. Otherwise we may, if necessary, add a root s to H (as we did in the proof of Theorem 10.6.4) without introducing any new directed cycles. Then Theorem 10.6.11 may be applied to the new graph. The values F k (v) can be calculated recursively using the initial values

and the identity

this obviously takes O(|V||E|) steps. After we have calculated all the F k (v), we may determine μ(H,w) with O(|V|2) comparisons, by Theorem 10.6.10. As H is connected, |V| is dominated by |E|, and the assertion follows. □

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