A Polyhedral Approach to Planar Augmentation and Related Problems

Given a planar graph G the planar biconnectivity augmen tation problem is to add the minimum number of edges to G such that the resulting graph is still planar and biconnected Given a nonplanar and biconnected graph the maximum planar biconnected subgraph problem consists of removing the minimum number of edges so that planarity is achieved and biconnectivity is maintained Both problems are important in Automatic Graph Drawing In JM the minimum planarizing k augmentation problem has been introduced that links the planarization step and the augmentation step together Here we are given a graph which is not necessarily planar and not necessarily k connected and we want to delete some set of edges D and to add some set of edges A such that jDj jAj is minimized and the resulting graph is planar k connected and spanning For all three problems we have given a poly hedral formulation by de ning three di erent linear objective functions over the same polytope namely the node connected planar spanning subgraph polytope NCPLS Kn We investigate the facial structure of this polytope for k which we will make use of in a branch and cut algorithm Here we give the dimension of the planar biconnected spanning subgraph polytope for G Kn and we show that all facets of the planar subgraph polytope PLS Kn are also facets of the new polytope NCPLS Kn Furthermore we show that the node cut con straints arising in the biconnectivity spanning subgraph polytope are facet de ning inequalities for NCPLS Kn We give rst computa tional results for all three problems the planar augmentation problem the minimum planarizing augmentation problem and the maximum planar biconnected spanning subgraph problem This is the rst time that instances of any of these three problems can be solved to optimality