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Unformatted text preview: 9.5 ArcDisjoint Branchings 503 above was two. BangJensen, Frank and Jackson proved that, if λ ( z,x ) ≥ k holds for those vertices x ∈ V ( D ) for which d + ( x ) > d ( x ) (that is, the value of k is restricted by the local arcconnectivities from z to these vertices), then a generalization is indeed possible. Theorem 9.5.2 [53] Let D = ( V,A ) be a directed multigraph with a spe cial vertex z and let T := { x ∈ V z : d ( x ) < d + ( x ) } . If λ ( z,x ) ≥ k ( ≥ 1) for every x ∈ T , then there is a family F of k arcdisjoint out arborescences rooted at z so that every vertex x ∈ V belongs to at least r ( x ) := min( k,λ ( z,x )) members of F . ut Clearly, if in Theorem 9.5.2 λ ( z,x ) ≥ k holds for every x ∈ V , then we are back at Edmonds’ theorem. Another special case is also worth mentioning. Call a directed multigraph D = ( V,A ) with root z a preflow directed multigraph if d ( x ) ≥ d + ( x ) holds for every x ∈ V z . (The name arises from a maxflow algorithms of Karzanov [475] and Goldberg and Tarjan [324], see also the definition of a preflow in Chapter 3). The following corollary of Theorem 9.5.2 may be considered as a generalization of Theorem 3.3.1. Corollary 9.5.3 [53] In a preflow directed multigraph D = ( V,A ) for any integer k ( ≥ 1) there is a family F of k arcdisjoint outarborescences with root z so that every vertex x belongs to min( k,λ ( z,x ; D )) members of F . In particular, if k := max( λ D ( z,x ) : x ∈ V z ) , then every x belongs to λ D ( z,x ) members of F . ut Aharoni and Thomassen have shown that Edmonds’ branching theorem cannot be generalized to infinite directed multigraphs [4]. 9.5.1 Implications of Edmonds’ Branching Theorem Below we give a number of nice consequences of Theorem 9.5.1 (for yet an other consequence see Theorem 9.7.2). The first result, due to Even, may be viewed as a generalization of Menger’s theorem for global arcstrong connec tivity. Corollary 9.5.4 [229, Theorem 6.10] Let D = ( V,A ) be a karcstrong di rected multigraph and let x,y be arbitrary distinct vertices of V . Then for every ≤ r ≤ k there exist paths P 1 ,P 2 ,...,P k in D which are arcdisjoint and such that the first r paths are ( x,y )paths and the last k r paths are ( y,x )paths. Proof: Let [ D,x,y ] be as described above. Add a new vertex s and join it to x by r parallel arcs of the form sx and to y by k r parallel arcs of the form sy . Let D denote the new directed multigraph. We claim that D satisfies (9.2). To see this let X ⊆ V be arbitrary. If X 6 = V , then we have d D ( X ) ≥ d D ( X ) ≥ k , since D is karcstrong. If X = V , we have 504 9. Disjoint Paths and Trees d D ( V ) = d + D ( s ) = k . It follows from Theorem 9.5.1 that D contains k arcdisjoint outbranchings all rooted at s . By the construction of D , these branchings restricted to D must consist of r outbranchings rooted at x and k r outbranchings rooted at y . Take the r ( x,y )paths from those rooted...
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 Spring '11
 Algorithms

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