Well-quasi-ordering of matrices under Schur complement and applications to directed graphs
European Journal of Combinatorics
Digraph width measures in parameterized algorithmics
Discrete Applied Mathematics
Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width
Fundamenta Informaticae - MFCS & CSL 2010 Satellite Workshops: Selected Papers
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A C-coloured graph is a graph, that is possibly directed, where the edges are coloured with colours from the set C. Clique-width is a complexity measure for C-coloured graphs, for finite sets C. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss some possible extensions of the notion of rank-width to C-coloured graphs. There is not a unique natural notion of rank-width for C-coloured graphs. We define two notions of rank-width for them, both based on a coding of C-coloured graphs by ${\mathbb{F}}^{*}$ -graphs-- $\mathbb {F}$ -coloured graphs where each edge has exactly one colour from $\mathbb{F}\setminus \{0\},\ \mathbb{F}$ a field--and named respectively $\mathbb{F}$ -rank-width and $\mathbb {F}$ -bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for $\mathbb{F}^{*}$ -graphs and prove that $\mathbb{F}^{*}$ -graphs of bounded $\mathbb{F}$ -rank-width are characterised by a list of $\mathbb{F}^{*}$ -graphs to exclude as vertex-minors (this list is finite if $\mathbb{F}$ is finite). An algorithm that decides in time O(n 3) whether an $\mathbb{F}^{*}$ -graph with n vertices has $\mathbb{F}$ -rank-width (resp. $\mathbb{F}$ -bi-rank-width) at most k, for fixed k and fixed finite field $\mathbb{F}$ , is also given. Graph operations to check MSOL-definable properties on $\mathbb{F}^{*}$ -graphs of bounded $\mathbb{F}$ -rank-width (resp. $\mathbb{F}$ -bi-rank-width) are presented. A specialisation of all these notions to graphs without edge colours is presented, which shows that our results generalise the ones in undirected graphs.