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Enumeration of Paths, Cycles, and Spanning Trees

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  • First Online:
Encyclopedia of Algorithms

Years and Authors of Summarized Original Work

  • 1975; Johnson

  • 1975; Read and Tarjan

  • 1994; Shioura, Tamura, Uno

  • 1995; Kapoor, Ramesh

  • 1999; Uno

  • 2013; Birmelé, Ferreira, Grossi, Marino, Pisanti, Rizzi, Sacomoto, Sagot

Problem Definition

Let G = (V, E) be a (directed or undirected) graph with n =  | V | vertices and m =  | E | edges. A walk of length k is a sequence of vertices v0, …, v k  ∈ V such that v i and vi+1 are connected by an edge of E, for any 0 ≤ i < k. A path π of length k is a walk v0, …, v k such that any two vertices v i and v j are distinct, for 0 ≤ i < j ≤ k: this is also called st-path where s = v0 and t = v k . A cycle (or, equivalently, elementary circuit) C of length k + 1 is a path v0, …, v k such that v k and v0 are connected by an edge of E.

We denote by \(\mathcal{P}_{st}(G)\) the set of st-paths in G for any two given vertices s, t ∈ V and by \(\mathcal{C}(G)\) the set of cycles in G. Given a graph G, the problem of st-path enumerationasks for generating all the...

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Authors and Affiliations

  1. Dipartimento di Informatica, Università di Pisa, Pisa, Italy

    Roberto Grossi

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  1. Roberto Grossi
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Correspondence to Roberto Grossi .

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Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Grossi, R. (2016). Enumeration of Paths, Cycles, and Spanning Trees. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_728

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