We call a digraph h-semicomplete if each vertex of the digraph has at most h non-neighbors, where a non-neighbor of a vertex v is a vertex u ≠ v such that there is no edge between u and v in either direction. This notion generalizes that of semicomplete digraphs which are 0-semicomplete and tournaments which are semicomplete and have no anti-parallel pairs of edges. Our results in this paper are as follows. (1) We give an algorithm which, given an h-semicomplete digraph G on n vertices and a positive integer k, in (h + 2k + 1)2knO(1) time either constructs a path-decomposition of G of width at most k or concludes correctly that the pathwidth of G is larger than k. (2) We show that there is a function f(k, h) such that every h-semicomplete digraph of pathwidth at least f(k, h) has a semicomplete subgraph of pathwidth at least k. One consequence of these results is that the problem of deciding if a fixed digraph H is topologically contained in a given h-semicomplete digraph G admits a polynomial-time algorithm for fixed h.
CITATION STYLE
Kitsunai, K., Kobayashi, Y., & Tamaki, H. (2015). On the pathwidth of almost semicomplete digraphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9294, pp. 816–827). Springer Verlag. https://doi.org/10.1007/978-3-662-48350-3_68
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