Chapter 3: Optimal Trees and Branchings - UKP

Optimal Trees and Branchings
Critical Graphs (6)
Corollary 16
For a branching B and a cycle C the following holds: either an edge e i ∈ C is
feasible w.r.t. B, or B contains all edges of C but one.
Theorem 17
Let H be a critical graph. Then, there is a branching B of maximal weight,
such that for any cycle C ⊆ H holds that |C \ B| = 1.
Proof. Let B be a maximum branching that contains as many edges as possible from H.
Consider a critical edge e ∈ H \ B:
If there is no e ′ ∈ B with t(e) = t(e ′ ), we could increase the number of edges from H in B
by simply adding e. This yields a branching B ′ containing more edges of H than B. Then
either B was not maximal, or B ′ contains a circuit.
If there is an e ′ ∈ B with t(e) = t(e ′ ): if e is feasible, exchanging e ′ by e would deliver a
maximum branching B ′ with more edges from H than B has. This contradicts the above
construction of B. It follows that there is no feasible edge e ∈ H \ B.
In particular, for any cycle C ⊆ H, no edge e ∈ C \ B is feasible. With the above Lemma, it
follows |C \ B| = 1 for any cycle C in H.
Efficient Graph Algorithms | Wolfgang Stille | WS 2011/2012 | Chapter III - Optimal Trees and Branchings | 50