Algorithmic aspects of graph connectivity by Hiroshi Nagamochi

By Hiroshi Nagamochi

Algorithmic features of Graph Connectivity is the 1st entire e-book in this critical proposal in graph and community thought, emphasizing its algorithmic elements. as a result of its large purposes within the fields of verbal exchange, transportation, and construction, graph connectivity has made great algorithmic development below the effect of the idea of complexity and algorithms in glossy desktop technology. The publication comprises a number of definitions of connectivity, together with edge-connectivity and vertex-connectivity, and their ramifications, in addition to comparable issues akin to flows and cuts. The authors comprehensively speak about new options and algorithms that let for speedier and extra effective computing, similar to greatest adjacency ordering of vertices. masking either easy definitions and complex issues, this ebook can be utilized as a textbook in graduate classes in mathematical sciences, similar to discrete arithmetic, combinatorics, and operations learn, and as a reference publication for experts in discrete arithmetic and its functions.

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15 ([273]). Let G, f , and Gˆ f be as defined earlier. Given a subset X ⊂ V such that s ∈ X and t ∈ V − X , a cut (X, V − X ) is a minimum (s, t)-cut in G if and only if ( Xˆ , Vˆ − Xˆ ) is a dicut in Gˆ f , where Xˆ is the set of vertices into which the vertices in X are contracted. The digraph Gˆ f is called a directed acyclic graph (DAG) representation of all minimum (s, t)-cuts in G. 18. The DAG representation for all minimum (s, t)-cuts in the digraph G of Fig. 14 with s = v1 and t = v12 .

This D ∗ is constructed so that a set of α-independent (s, t)-paths in G corresponds to a set of the same number of edge-disjoint (s, t)-paths in D ∗ , and vice versa. , that of α-independent (s, t)-paths in G) is equal to the size of a minimum (s, t)-cut X ∗ in D ∗ . By letting Z = {v ∈ V − {s, t} | (v , v ) ∈ E(X ∗ ; D ∗ )} and A = {u ∈ V − Z | (u , v ) ∈ E(X ∗ ; D ∗ )}, we see that (A, B = V − Z − A, Z ) is a mixed cut separating s and t, and it has the size α(Z ) + d(A, B; G) = d(X ∗ ; D ∗ ).

We now explain the basic ideas of such an algorithm. Given an (s, t)-cut X in a digraph G = (V, E), we say that two vertices u, v ∈ V are separated by X if |X ∩ {u, v}| = 1 holds. Let f be a maximum (s, t)-flow and G f be its residual graph. 14) holds. Therefore, for any directed cycle C in the residual graph G f , the end vertices u and v of an edge (u, v) in C are not separated in G by any minimum (s, t)-cut. From this we see that all minimum (s, t)-cuts in G are preserved after contracting each strongly connected component of G f into a single vertex.

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