Common-Face Embeddings of Planar Graphs

TL;DR

This paper studies a complex graph embedding problem with face constraints, proving NP-completeness generally but providing an efficient solution for a special connected subgraph case, relevant to geographic and VLSI applications.

Contribution

It introduces a new face embedding problem with constraints, proves its NP-completeness, and offers a polynomial-time solution for cases with connected subgraph constraints.

Findings

NP-complete in general case

Efficient O(I log I) algorithm for connected subgraph case

Classical planar embedding is a special case

Abstract

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.