Non-planar Core Reduction of Graphs

Abstract

We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to planarity measures like crossing number, skewness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be computed in linear time. It has applications in heuristic and exact optimization algorithms for the planarity measures mentioned above. Experimental results show that this strategy yields a reduction to 2/3 in average for a widely used benchmark set of graphs.

Experiments on exact crossing minimization using column generation

“…But even for 2-connected graphs, further reductions are possible. The algorithm presented in Gutwenger and Chimani [2005] is based on the following fact: Suppose G is the union of two edge-disjoint graphs S and K sharing exactly two vertices s and t, and the graph S ∪ (s, t) is planar; such a subgraph S of G is also called a planar 2-component of G. Then, there exists a drawing D S of S with both s and t on the external face. Moreover, it is possible to draw a closed curve in D S that crosses exactly the edges of a minimum st-cut in S. It turns out that the crossing number of G equals the (weighted) crossing number of the graph K * := K ∪ (s, t) in which the additional edge e st := (s, t) has a weight of λ.…”

“…Note that even our pure branch-and-cut algorithm is already an improved version of the one presented in Buchheim et al [2005], due to the use of primal heuristics during the cutting phase, as well as the recently developed preprocessing method called nonplanar core [Gutwenger and Chimani 2005]. The dataset used for the study is the well known Rome library [Di Battista et al 1997], which has, for example, also been used for a comparison between crossing minimization heuristics [Gutwenger and Mutzel 2004].…”