# Directed Path-Width of Sequence Digraphs

## Abstract

Computing the directed path-width of a digraph is NP-hard even for digraphs of maximum semi-degree 3. In this paper we consider a family of graph classes called sequence digraphs, such that for each of these classes the directed path-width can be computed in polynomial time. For this purpose we define the graph classes \(S_{k,\ell }\) as the set of all digraphs \(G=(V,A)\) which can be defined by *k* sequences with at most \(\ell \) entries from *V*, such that \((u,v) \in A\) if and only if in one of the sequences *u* occurs before *v*. We characterize digraphs which can be defined by \(k=1\) sequence by four forbidden subdigraphs and also as a subclass of semicomplete digraphs. Given a decomposition of a digraph *G* into *k* sequences, we show an algorithm which computes the directed path-width of *G* in time Open image in new window , where *N* denotes the maximum sequence length. This leads to an XP-algorithm w.r.t. *k* for the directed path-width problem. As most known parameterized algorithms for directed path-width consider the standard parameter, our algorithm improves significantly the known results for a high amount of digraphs of large directed path-width.

## Keywords

Directed path-width Transitive tournament Semicomplete digraph XP-algorithm## References

- 1.Bang-Jensen, J., Gutin, G.: Digraphs. Theory, Algorithms and Applications. Springer, Berlin (2009). https://doi.org/10.1007/978-1-84800-998-1CrossRefzbMATHGoogle Scholar
- 2.Barát, J.: Directed pathwidth and monotonicity in digraph searching. Graphs Combin.
**22**, 161–172 (2006)MathSciNetCrossRefGoogle Scholar - 3.Bodlaender, H.L.: A partial \(k\)-arboretum of graphs with bounded treewidth. Theor. Comput. Sci.
**209**, 1–45 (1998)MathSciNetCrossRefGoogle Scholar - 4.Gould, R.: Graph Theory. Dover, Downers Grove (2012)zbMATHGoogle Scholar
- 5.Gurski, F., Rethmann, J., Wanke, E.: On the complexity of the FIFO stack-up problem. Math. Methods Oper. Res.
**83**(1), 33–52 (2016)MathSciNetCrossRefGoogle Scholar - 6.Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. J. Combin. Theory Ser. B
**82**, 138–155 (2001)MathSciNetCrossRefGoogle Scholar - 7.Kitsunai, K., Kobayashi, Y., Komuro, K., Tamaki, H., Tano, T.: Computing directed pathwidth in \({O}(1.89^n)\) time. Algorithmica
**75**, 138–157 (2016)MathSciNetCrossRefGoogle Scholar - 8.Kitsunai, K., Kobayashi, Y., Tamaki, H.: On the pathwidth of almost semicomplete digraphs. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 816–827. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48350-3_68CrossRefGoogle Scholar
- 9.Kobayashi, Y.: Computing the pathwidth of directed graphs with small vertex cover. Inf. Process. Lett.
**115**(2), 310–312 (2015)MathSciNetCrossRefGoogle Scholar - 10.Monien, B., Sudborough, I.H.: Min cut is NP-complete for edge weighted trees. Theor. Comput. Sci.
**58**, 209–229 (1988)MathSciNetCrossRefGoogle Scholar - 11.Nagamochi, H.: Linear layouts in submodular systems. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 475–484. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-35261-4_50CrossRefGoogle Scholar