Common-Face Embeddings of Planar Graphs
<|reference_start|>Common-Face Embeddings of Planar Graphs: Given a planar graph G and a sequence C_1,...,C_q, where each C_i 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 in {1,...,q}, there is a face F_i in the embedding whose boundary contains at least one vertex from each set in C_i. 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 C_i, 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 C_i, each set in C_i 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 C_1,...,C_q only incurs a logarithmic factor of overhead.<|reference_end|>
@article{chen2001common-face,
    title={Common-Face Embeddings of Planar Graphs},
    author={Zhi-Zhong Chen, Xin He, Ming-Yang Kao},
    journal={arXiv preprint arXiv:cs/0102007},
    year={2001},
    archivePrefix={arXiv},
    eprint={cs/0102007},
    primaryClass={cs.DS cs.DM}
}