{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Digraphs+Theory,+Algorithms+and+Applications_Part20

# Digraphs+Theory,+Algorithms+and+Applications_Part20 - 7.6...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 7.6 Increasing the Arc-Strong 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 k-arc-strong). 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 arc-strong con- nectivity . By Menger’s theorem, D is k-arc-strong if and only if γ k ( D ) ≤ 0. Indeed, if D is k-arc-strong, 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 k-arc- 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 k-out-critical 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 k-out-critical 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 )....
View Full Document

{[ snackBarMessage ]}

### Page1 / 20

Digraphs+Theory,+Algorithms+and+Applications_Part20 - 7.6...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online