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Are there any good digraph width measures? (English) Zbl 1327.05136
Summary: Many width measures for directed graphs have been proposed in the last few years in pursuit of generalizing (the notion of) treewidth to directed graphs. However, none of these measures possesses, at the same time, the major properties of treewidth, namely, (1) being algorithmically useful, that is, admitting polynomial-time algorithms for a large class of problems on digraphs of bounded width (e.g. the problems definable in $$\mathrm{MSO}_1$$); (2) having nice structural properties such as being (at least nearly) monotone under taking subdigraphs and some form of arc contractions (property closely related to characterizability by particular cops-and-robber games). We investigate the question whether the search for directed treewidth counterparts has been unsuccessful by accident, or whether it has been doomed to fail from the beginning. Our main result states that any reasonable width measure for directed graphs which satisfies the two properties above must necessarily be similar to treewidth of the underlying undirected graph.

MSC:
 05C20 Directed graphs (digraphs), tournaments 05C12 Distance in graphs 05C05 Trees 91A24 Positional games (pursuit and evasion, etc.)
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