The Polyhedral Approach to the Maximum Planar Subgraph Problem: New Chances for Related Problems

In [JM94] 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 2 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..