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Given a planar graph $\Ggg$ and a sequence ${\CC}_1,\ldots,{\CC}_q$, where each ${\CC}_i$ is a family of vertex subsets of $\Ggg$, we wish to find a plane embedding of $\Ggg$, if any exists, such that, for each $i\in\{1,\ldots,q\}$, there is a face Fi in the embedding whose boundary contains at least one vertex from each set in CCi. This problem has applications in the recovery of topological information from geographical data and the design of constrained layouts in VLSI. Let $\inputsize$ be the input size, i.e., the total number of vertices and edges in $\Ggg$ and the families CCi, counting multiplicity. We show that this problem is NP-complete in general. We also show that it is solvable in $O(\inputsize\log \inputsize)$ time for the special case in which, for each input family CCi, each set in CCi induces a connected subgraph of the input graph $\Ggg$. Note that the classical problem of simply finding a planar embedding is a further special case of this case with q=0. Therefore, the processing of the additional constraints CC1, . . .,CCq incurs only a logarithmic factor of overhead.

Common-Face Embeddings of Planar Graphs