Optimizing over all combinatorial embeddings of a planar graph

Optimizng Over All Combinatorial Embeddings of a Planar Graph". We study the problem of optimizing over the set of all combinatorial embeddings of a given planar graph. Our objective function prefers certain cycles of $G$ as face cycles in the embedding. The motivation for studying this problem arises in graph drawing, where the chosen embedding has an important influence on the aesthetics of the drawing. We characterize the set of all possible embeddings of a given biconnected planar graph $G$ by means of a system of linear inequalities with ${0,1}$-variables corresponding to the set of those cycles in $G$ which can appear in a combinatorial embedding. This system of linear inequalities can be constructed recursively using the data structure of SPQR-trees and a new splitting operation. Our computational results on two benchmark sets of graphs are surprising: The number of variables and constraints seems to grow only linearly with the size of the graphs although the number of embeddings grows exponentially. For all tested graphs (up to 500 vertices) and linear objective functions, the resulting integer linear programs could be generated within 600 seconds and solved within two seconds on a Sun Enterprise 10000 using CPLEX