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**1 - 4**of**4**### Common-Face Embeddings of Planar Graphs

, 2001

"... Given a planar graph G and a sequence C1,...,Cq, where each Ci is a family of vertex subsets of G, we wish to find a plane embedding of G, if any exists, such that for each i ∈ {1,..., q}, there is a face Fi in the embedding whose boundary contains at least one vertex from each set in Ci. This probl ..."

Abstract
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Given a planar graph G and a sequence C1,...,Cq, where each Ci is a family of vertex subsets of G, we wish to find a plane embedding of G, if any exists, such that for each i ∈ {1,..., q}, there is a face Fi in the embedding whose boundary contains at least one vertex from each set in Ci. This problem has applications to the recovery of topological information from geographical data and the design of constrained layouts in VLSI. Let I be the input size, i.e., the total number of vertices and edges in G and the families Ci, counting multiplicity. We show that this problem is NP-complete in general. We also show that it is solvable in O(I log I) time for the special case where for each input family Ci, each set in Ci induces a connected subgraph of the input graph G. 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 C1,..., Cq only incurs a logarithmic factor of overhead.