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.
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.
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.
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.
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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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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