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Forbidden Directed Minors, Directed Path-Width and Directed Tree-Width of Tree-Like Digraphs

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Book cover SOFSEM 2019: Theory and Practice of Computer Science (SOFSEM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11376))

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Abstract

There have been many attempts to find directed graph classes with bounded directed path-width and bounded directed tree-width. Right now, the only known directed tree-width-/path-width-bounded graphs are cycle-free graphs with directed path-width and directed tree-width 0. In this paper, we introduce directed versions of cactus trees and pseudotrees and -forests and characterize them by at most three forbidden directed graph minors. Furthermore, we show that directed cactus trees and forests have a directed tree-width of at most 1 and directed pseudotrees and -forests even have a directed path-width of at most 1.

The work of the second author was supported by the German Research Association (DFG) grant GU 970/7-1.

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Notes

  1. 1.

    This means, in graph \(G'\) the edge e and its two incident vertices u and v are replaced by the vertex w and all other edges in G incident with u or v are adjacent with w in \(G'\).

  2. 2.

    This means, in digraph \(G'\) the edge e and its two incident vertices u and v are replaced by the vertex w and all other edges in G incident with u or v are incident with w in \(G'\).

  3. 3.

    This means, in digraph \(G'\) the cycle C is replaced by the vertex w and all other edges in G incident with a vertex in C are incident with w in \(G'\).

  4. 4.

    A remarkable difference to the undirected tree-width [13] is that the sets \(W_r\) have to be disjoint and non-empty.

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Author information

Authors and Affiliations

  1. Institute of Computer Science, Algorithmics for Hard Problems Group, Heinrich-Heine-University Düsseldorf, 40225, Düsseldorf, Germany

    Frank Gurski & Carolin Rehs

Authors
  1. Frank Gurski
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  2. Carolin Rehs
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Corresponding author

Correspondence to Carolin Rehs .

Editor information

Editors and Affiliations

  1. University of Genoa, Genoa, Italy

    Barbara Catania

  2. Comenius University, Bratislava, Slovakia

    Rastislav Královič

  3. Poznań University of Technology, Poznań, Poland

    Jerzy Nawrocki

  4. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

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Gurski, F., Rehs, C. (2019). Forbidden Directed Minors, Directed Path-Width and Directed Tree-Width of Tree-Like Digraphs. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_19

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  • DOI: https://doi.org/10.1007/978-3-030-10801-4_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10800-7

  • Online ISBN: 978-3-030-10801-4

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