By Hiroshi Nagamochi

ISBN-10: 0521878640

ISBN-13: 9780521878647

Algorithmic features of Graph Connectivity is the 1st finished ebook in this significant proposal in graph and community concept, emphasizing its algorithmic features. due to its vast purposes within the fields of verbal exchange, transportation, and construction, graph connectivity has made great algorithmic development lower than the impact of the idea of complexity and algorithms in smooth laptop technological know-how. The ebook comprises a number of definitions of connectivity, together with edge-connectivity and vertex-connectivity, and their ramifications, in addition to similar subject matters comparable to flows and cuts. The authors comprehensively speak about new recommendations and algorithms that let for speedier and extra effective computing, resembling greatest adjacency ordering of vertices. protecting either simple definitions and complicated issues, this booklet can be utilized as a textbook in graduate classes in mathematical sciences, corresponding to discrete arithmetic, combinatorics, and operations learn, and as a reference ebook for experts in discrete arithmetic and its functions.

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**Extra info for Algorithmic aspects of graph connectivity**

**Example text**

A problem is usually described as a mathematical statement that contains several parameters; a problem instance is obtained by assigning values to those parameters. Thus, a problem can be viewed as a collection of (usually infinitely many) such instances. ” An optimization problem asks for a solution that minimizes (or maximizes) a given objective function among all feasible solutions. The class P, which stands for “polynomial,” consists of all decision problems that admit polynomial time algorithms.

For an undirected (resp. 21. The augmented digraph G S for a digraph G and a subset S ⊆ V . E(L , L) (resp. (u, v) ∈ E(L , L)) (note that L does not necessarily induce a kvertex-connected subgraph from G). By definition, any subset L ⊆ V is κ L ,L vertex-connected, and, for any vertex cut C with |C| < κ L ,L , S − C is contained in the same component (resp. strongly connected component) in G − C, since κ(u, v; G − C) ≥ κ L ,L − |C| ≥ 1 holds for all {u, v} ∈ E(L − C, L − C). The next property gives a condition by which we can omit computing κT,T to determine κ(G).

D(X ; G ) = d(X ; G )) holds. Since G is d(X ; G s,t ) = d(X ; G ) + d(V − X ; G ) = 2d(X ; G ). Therefore, d(X ; G) = d(X ; G s,t )/2 − max{ , 0}. In particular, an (s, t)-cut X is minimum in G if and only if it also is in G s,t . Clearly G s,t can be obtained from G in O(n + m) time, and it has at most n more edges than the original digraph G. 1 Menger’s Theorem Menger’s theorem [217] states that the maximum number of edge-disjoint (resp. internally vertex-disjoint) (s, t)-paths is equal to the minimum size of an (s, t)-cut (resp.

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