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Testing C-planarity of Embedded Graphs

We show that c-planarity is solvable in a polynomial time for flat clustered graphs with three clusters if the combinatorial embedding of the abstract graph is fixed. In other words, given a graph G embedded in the plane whose vertices are partition into three parts our algorithm decides if there exists a plane supergraph G' of G on the same vertex set in which the vertices of each part induce a connected sub-graph contained in the outer-face of its complement in G'. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic cluster graphs.