Solving the Maximum Weight Planar Subgraph Problem by Branch-and-Cut

In this paper we investigate the problem of identifying a planar subgraph of maximum weight of a given edge weighted graph. In the theoretical part of the paper, the polytope of all planar subgraphs of a graph G is defined and studied. All subgraphs of a graph G, which are subdivisions of K5 or K3,3, turn out to define facets of this polytope. 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 K5 or K3,3. These structures give us inequalities which are used as cutting planes