Digraphs+Theory,+Algorithms+and+Applications_Part27

Digraphs+Theory,+Algorithms+and+Applications_Part27 - 9.5...

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Unformatted text preview: 9.5 Arc-Disjoint Branchings 503 above was two. Bang-Jensen, Frank and Jackson proved that, if Î» ( z,x ) â‰¥ k holds for those vertices x âˆˆ V ( D ) for which d + ( x ) > d- ( x ) (that is, the value of k is restricted by the local arc-connectivities 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 âˆˆ V- z : d- ( x ) < d + ( x ) } . If Î» ( z,x ) â‰¥ k ( â‰¥ 1) for every x âˆˆ T , then there is a family F of k arc-disjoint out- arborescences rooted at z so that every vertex x âˆˆ V belongs to at least r ( x ) := min( k,Î» ( z,x )) members of F . ut Clearly, if in Theorem 9.5.2 Î» ( z,x ) â‰¥ k holds for every x âˆˆ V , then we are back at Edmondsâ€™ 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 ) â‰¥ d + ( x ) holds for every x âˆˆ V- z . (The name arises from a max-flow 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 ( â‰¥ 1) there is a family F of k arc-disjoint out-arborescences with root z so that every vertex x belongs to min( k,Î» ( z,x ; D )) members of F . In particular, if k := max( Î» D ( z,x ) : x âˆˆ V- z ) , then every x belongs to Î» D ( z,x ) members of F . ut Aharoni and Thomassen have shown that Edmondsâ€™ branching theorem cannot be generalized to infinite directed multigraphs [4]. 9.5.1 Implications of Edmondsâ€™ 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 Mengerâ€™s theorem for global arc-strong connec- tivity. Corollary 9.5.4 [229, Theorem 6.10] Let D = ( V,A ) be a k-arc-strong di- rected multigraph and let x,y be arbitrary distinct vertices of V . Then for every â‰¤ r â‰¤ k there exist paths P 1 ,P 2 ,...,P k in D which are arc-disjoint 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 âŠ† V be arbitrary. If X 6 = V , then we have d- D ( X ) â‰¥ d- D ( X ) â‰¥ k , since D is k-arc-strong. 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 arc-disjoint out-branchings all rooted at s . By the construction of D , these branchings restricted to D must consist of r out-branchings rooted at x and k- r out-branchings rooted at y . Take the r ( x,y )-paths from those rooted...
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Digraphs+Theory,+Algorithms+and+Applications_Part27 - 9.5...

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