Characterizations and Directed Path-Width of Sequence Digraphs

Computing the directed path-width of a directed graph is an NP-hard problem. Even for digraphs of maximum semi-degree 3 the problem remains hard. We propose a decomposition of an input digraph G = ( V , A ) by a number k of sequences with entries from V , such that ( u , v ) ∈ A if and only if in one of the sequences there is an occurrence of u appearing before an occurrence of v . We present several graph theoretical properties of these digraphs. Among these we give forbidden subdigraphs of digraphs which can be defined by k = 1 sequence, which is a subclass of semicomplete digraphs. Given the decomposition of digraph G , we show an algorithm which computes the directed path-width of G in time 𝒪(k· (1+N)^k) , where N denotes the maximum sequence length. This leads to an XP-algorithm w.r.t. k for the directed path-width problem. Our result improves the algorithms of Kitsunai et al. for digraphs of large directed path-width which can be decomposed by a small number of sequences and confirm their conjecture that semicompleteness is a useful restriction when considering digraphs.