On the Minimum Cut of Planarizations

Markus Chimani, Carsten Gutwenger, Petra Mutzel
2007 Electronic Notes in Discrete Mathematics  
Every drawing of a non-planar graph G in the plane induces a planarization, i.e., a planar graph obtained by replacing edge crossings with dummy vertices. In this paper, we consider the relationship between the capacity of a minimum st-cut in a graph G and its planarizations. We show that these capacities need not be equal. On the other hand, we prove that every crossing minimal planarization can be efficiently transformed into another crossing minimal planarization that preserves the capacity
more » ... f a minimum st-cut in G. Furthermore, we extend the result to general (reasonable) planarizations. This property turns out to be a powerful tool for reducing the computational efforts in crossing minimization algorithms. Another application is the correction of a proof given byŠiráň [8] , that shows an additivity property of the crossing number with respect to certain decompositions.
doi:10.1016/j.endm.2007.01.036 fatcat:cxsebbtjlne2rkhnacydltmizi