# The polyhedral approach to the maximum planar subgraph problem: New chances for related problems

## Abstract

- (1)
Find a maximum planar subgraph with maximum degree

*d*∈ 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 [JM93], the *k*-connected subgraph polytope [S92] 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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