The Polyhedral Approach to the Maximum Planar Subgraph Problem: New Chances for Related Problems
Jünger, Michael and Mutzel, Petra (1994) The Polyhedral Approach to the Maximum Planar Subgraph Problem: New Chances for Related Problems. [Preprint]
In Michael Jünger and Petra Mutzel [Algorithmica, 16 (1996)] we used a branch-and-cut algorithm in order to determine a maximum weight planar subgraph of a given graph. One of the motivations was to produce a nice drawing of a given graph by drawing the found maximum planar subgraph, and then augmenting this drawing by the removed edges. Our experiments indicate that drawing algorithms for planar graphs which require 2- or 3-connectivity, resp. degree-constraints, in addition to planarity often give ''nicer'' results. Thus we are led to the following problems: 1. Find a maximum planar subgraph with maximum degree d in IN. 2. Augment a planar graph to a k-connected planar graph. 3. Find a maximum planar k-connected subgraph of a given k-connected graph. 4. Given a graph G, which is not necessarily planar and not necessarily k-connected, determine a new graph H by removing r edges and adding a edges such that the new graph H is planar, spanning, k-connected, each node v has degree at most D(v) and r+a is minimum. Problems (1), (2) and (3) have been discussed in the literature, we argue that a solution to the newly defined problem (4) is most useful for our goal. For all four problems we give a polyhedral formulation by defining different linear objective functions over the same polytope which is the intersection of the planar subgraph polytope, see Michael Jünger and Petra Mutzel [Proc. IPCO3 (1993)], the k-connected subgraph polytope M. Stoer [LNCS 1531 (1992)] and the degree-constrained subgraph polytope. We point out why we are confident that a branch-and-cut algorithm for the new problem will be an implementable and useful tool in automatic graph drawing.
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