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Quasi-Transitive Digraphs and Their Extensions

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Classes of Directed Graphs

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Abstract

A digraph D is quasi-transitive if for any three distinct vertices x, y, z in D, the existence of the arcs xy and yz in D implies that xz, zx or both are arcs of D. Quasi-transitive digraphs generalize both tournaments (and semicomplete digraphs) and transitive digraphs, and share some of the nice properties of these families. In particular, many problems that are \(\mathcal {NP}\)-complete for general digraphs become solvable in polynomial time when restricted to quasi-transitive digraphs. In this chapter, we focus on presenting how usually difficult problems admit efficient solutions for the family of quasi-transitive digraphs and some of its generalizations. We begin with the study of the structure of quasi-transitive digraphs, given by the recursive characterization theorem known as the Canonical Decomposition Theorem; two generalizations of quasi-transitive digraphs are introduced. We define a digraph D to be k-quasi-transitive if for any pair of vertices x, y in D, the existence of a path of length k from x to y implies that xy, yx or both are arcs of D. Given a class of digraphs \(\varPhi \), we say that a digraph is totally \(\varPhi \)-decomposable if it can be expressed as a composition of totally \(\varPhi \)-decomposable digraphs; this concept generalizes the structure of quasi-transitive digraphs given by the Canonical Decomposition Theorem. Some of the problems studied for quasi-transitive digraphs and its generalizations include hamiltonicity, traceability, k-linkages weak k-linkages, existence and number of k-kings, the Path Partition Conjecture and pancyclicity. A brief section is devoted to homomorphisms in transitive digraphs.

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Notes

  1. 1.

    He proved that a graph G admits a quasi-transitive orientation if and only if it admits a transitive orientation if and only if it is a comparability graph.

  2. 2.

    Contraction is defined in Section 1.4 for directed multigraphs. We can obtain a digraph instead of a directed multigraph by deleting spare parallel arcs after contraction.

  3. 3.

    Recall that the g.c.d. of two integers is their least positive linear combination. Clearly, 4 is a linear combination of \(k-1\) and \(k+3\), but since k is even, and \(k-1 \not \equiv k+3\) (mod 3), the least positive linear combination of \(k-1\) and \(k+3\) is 1.

  4. 4.

    Sometimes we allow that the paths may share one or both of their end-vertices, i.e., \(V(P_i)\cap V(P_j) \subseteq \{x_i,y_i,x_j,y_j\}\) whenever \(i \ne j\), where \(x_i=y_j\) or \(x_i=x_j\) is possible.

  5. 5.

    Note that the same pair (or the same vertex) may appear more than once in the list and we may have \(s_i=t_i\).

  6. 6.

    Note that an external path may still start and end in the same module \(H_j\).

  7. 7.

    Note that the running time of \(\mathcal B_{\varPhi }\) may depend heavily on c.

  8. 8.

    \(K_i [ W_i ]\) is the subdigraph of \(K_i\) induced by \(W_i\).

  9. 9.

    See Section 1.4.

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Área de la Investigación Científica, Circuito Exterior, Ciudad Universitaria, C.P., 045010, Coyoacán, México, CDMX, México

    Hortensia Galeana-Sánchez & César Hernández-Cruz

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  1. Hortensia Galeana-Sánchez
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  2. César Hernández-Cruz
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Correspondence to César Hernández-Cruz .

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Editors and Affiliations

  1. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Jørgen Bang-Jensen

  2. Department of Computer Science, Royal Holloway, University of London, Egham, United Kingdom

    Gregory Gutin

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Galeana-Sánchez, H., Hernández-Cruz, C. (2018). Quasi-Transitive Digraphs and Their Extensions. In: Bang-Jensen, J., Gutin, G. (eds) Classes of Directed Graphs. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-71840-8_8

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