Maximum Planar Subgraphs and Nice Embeddings :

In automatic graph drawing a given graph has to be layed-out in the plane, usually according to a number of topological and aesthetic constraints. Nice drawings for sparse nonplanar graphs can be achieved by determining a maximum planar subgraph and augmenting an embedding of this graph. This approach appears to be of limited value in practice, because the maximum planar subgraph problem is NP-hard. We attack the maximum planar subgraph problem with a branch and cut technique which gives us quite good and in many cases provably optimum solutions for sparse graphs and very dense graphs. In the theoretical part of the paper, the polytope of all planar subgraphs of a graph G is deened and studied. All subgraphs of a graph G, which are subdivisions of K 5 or K 3;3 , turn out to deene facets of this polytope. For cliques contained in G, the Euler inequalities turn out to be facet-deening for the planar subgraph polytope. Moreover we introduce the subdivision inequalities, V 2k inequalities and the ower inequalities all of which are facet-deening for the polytope. Furthermore, the composition of inequalities by 2-sums is investigated. We also present computational experience with a branch and cut algorithm for the above problem. Our approach is based on an algorithm which searches for forbidden substructures in a graph that contains a subdivision of K 5 or K 3;3. These structures give us inequalities which are used as cutting planes. Finally, we try to convince the reader that the computation of maximum planar subgraphs is indeed a practical tool for nding nice embeddings by applying this method to graphs taken from the literature.