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Unformatted text preview: 7.6 Increasing the ArcStrong Connectivity Optimally 363 arcs. First let us make the simple observation that such a set F indeed exists, since we may just add k parallel arcs in both directions between a fixed vertex v ∈ V and all other vertices in V (it is easy to see that the resulting directed multigraph will be karcstrong). Definition 7.6.1 Let D = ( V,A ) be a directed multigraph. Then γ k ( D ) is the smallest integer γ such that X X i ∈F ( k d ( X i )) ≤ γ and X X i ∈F ( k d + ( X i )) ≤ γ, for every subpartition F = { X 1 ,...,X t } of V with ∅ ⊂ X i ⊂ V , i = 1 ,...,t. We call γ k ( D ) the subpartition lower bound for arcstrong con nectivity . By Menger’s theorem, D is karcstrong if and only if γ k ( D ) ≤ 0. Indeed, if D is karcstrong, then d + ( X ) ,d ( X ) ≥ k holds for all proper subsets of V and hence we see that γ k ( D ) ≤ 0. Conversely, if D is not karc strong, then let X be a set with d ( X ) < k . Take F = { X } , then we see that γ k ( D ) ≥ k d ( X ) > 0. Lemma 7.6.2 [258] Let D = ( V,A ) be a directed multigraph and let k be a positive integer such that γ k ( D ) > . Then D can be extended to a new directed multigraph D = ( V + s,A ∪ F ) , where F consists of γ k ( D ) arcs whose head is s and γ k ( D ) arcs of whose tail is s such that (7.9) holds in D . Proof: We will show that, starting from D , it is possible to add γ k ( D ) arcs from V to s so that the resulting graph satisfies d + ( X ) ≥ k for all ∅ 6 = X ⊂ V. (7.13) Then it will follow analogously (by considering the converse of D ) that it is also possible to add γ k ( D ) new arcs from s to V so that the resulting graph satisfies d ( X ) ≥ k for all ∅ 6 = X ⊂ V. (7.14) First add k parallel arcs from v to s for every v ∈ V . This will certainly make the resulting directed multigraph satisfy (7.13). Now delete as many new arcs as possible until removing any further arc would result in a digraph where (7.13) no longer holds (that is, every remaining new arc vs leaves a koutcritical set). Let ˜ D denote the current directed multigraph after this deletion phase and let S be the set of vertices v which have an arc to s in ˜ D . Let F = { X 1 ,...,X r } be a family of koutcritical sets such that every v ∈ S is contained in some member X i of F and assume that F has as few members as possible with respect to this property. Clearly this choice implies that either F is a subpartition of V , or there is a pair of intersecting sets X i ,X j in F . 364 7. Global Connectivity Case 1: F is a subpartition of V . Then we have kr = r X i =1 d + ˜ D ( X i ) = r X i =1 ( d + D ( X i ) + d + ˜ D ( X i ,s )) = r X i =1 d + D ( X i ) + d ˜ D ( s ) , implying that d ˜ D ( s ) = ∑ r i =1 ( k d + D ( X i )) ≤ γ k ( D ), by the definition of γ k ( D )....
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 Spring '11
 Algorithms

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