\mathbb F\mathbb F-Rank-Width of (Edge-Colored) Graphs

Rank-width is a complexity measure equivalent to the clique-width of undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss an extension of the notion of rank-width to all types of graphs - directed or not, with edge colors or not -, named \(\mathbb F\)-rank-width. We extend most of the results known for the rank-width of undirected graphs to the \(\mathbb F\)-rank-width of graphs: cubic-time recognition algorithm, characterisation by excluded configurations under vertex-minor and pivot-minor, and algebraic characterisation by graph operations. We also show that the rank-width of undirected graphs is a special case of \(\mathbb F\)-rank-width.

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