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 \documentclass[12pt]{minimal}
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\begin{document}${\mathbb{F}}^{*}$\end{document}-graphs—\documentclass[12pt]{minimal}
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\begin{document}$\mathbb {F}$\end{document}-coloured graphs where each edge has exactly one colour from \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}\setminus \{0\},\ \mathbb{F}$\end{document} a field—and named respectively \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}$\end{document}-rank-width and \documentclass[12pt]{minimal}
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\begin{document}$\mathbb {F}$\end{document}-bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}^{*}$\end{document}-graphs and prove that \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}^{*}$\end{document}-graphs of bounded \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}$\end{document}-rank-width are characterised by a list of \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}^{*}$\end{document}-graphs to exclude as vertex-minors (this list is finite if \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}$\end{document} is finite). An algorithm that decides in time O(n3) whether an \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}^{*}$\end{document}-graph with n vertices has \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}$\end{document}-rank-width (resp. \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}$\end{document}-bi-rank-width) at most k, for fixed k and fixed finite field \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}$\end{document}, is also given. Graph operations to check MSOL-definable properties on \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}^{*}$\end{document}-graphs of bounded \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}$\end{document}-rank-width (resp. \documentclass[12pt]{minimal}
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\begin{document}$\mathbb{F}$\end{document}-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.
