Solving the Maximum Weight Planar

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 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. 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. A graph G = (V; E) is said to be planar, if it can be drawn on the plane such that no two edges intersect geometrically except at a vertex at which they are both incident. According to Kuratowski's Theorem, planar graphs are exactly the graphs that contain no subdivisions of K 5 or K 3;3. Given a nonplanar weighted graph with edge weights w e for e 2 E we want delete a set of edges F to obtain a planar subgraph G 0 = (V; E n F) such that the sum of all edge weights P e2EnF c e of G 0 is maximum. In the unweighted case, where c e = 1 for all edges e 2 E, the problem consists of nding the minimum number of edges whose deletion from a nonplanar graph gives a planar subgraph. In either case the problem is NP-hard GJ79]. The problem can be solved in polynomial time if G is already planar, since planarity testing can be done in linear time HT74]. If G = K n , the complete graph on n nodes, or G = K m;n , the complete bipartite graph on n + m nodes, it is easy to construct a solution which contains 3n ? 6, resp. 2n ? 4 edges, and so the unweighted problem is solved in linear time. A related problem to the unweighted maximum planar subgraph problem is the maximal planar subgraph problem. It consists of nding a planar subgraph G 0 = (V 0 ; E 0) such that for all edges e 2 E n E 0 the addition of e to G 0 destroys the planarity of G 0. Recently …