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Abstract

Given a planar graph $G$, the planar (biconnectivity) augmentation 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 [JM95], 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 $|D|+|A|$ is minimized and the resulting graph is planar, $k$-connected and spanning. For all three problems, we have given a polyhedral formulation by defining three different linear objective functions over the same polytope, namely the $2$-node connected planar spanning subgraph polytope $\2NCPLS(K_n)$. We investigate the facial structure of this polytope for $k=2$, 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 $G=K_n$ and we show that all facets of the planar subgraph polytope $\PLS(K_n)$ are also facets of the new polytope $\2NCPLS(K_n)$. Furthermore, we show that the node-cut constraints arising in the biconnectivity spanning subgraph polytope, are facet-defining inequalities for $\2NCPLS(K_n)$. We give first computational results for all three problems, the planar $2$-augmentation problem, the minimum planarizing $2$-augmentation problem and the maximum planar biconnected (spanning) subgraph problem. This is the first time that instances of any of these three problems can be solved to optimality.