Planarization

In graph drawing, planarization is a method of extending drawing methods from planar graphs to graphs that are not planar, by embedding the non-planar graphs within a larger planar graph.

Planarization may be performed by using any method to find a drawing (with crossings) for the given graph, and then replacing each crossing point by a new artificial vertex, causing each crossed edge to be subdivided into a path. The original graph will be represented as a immersion minor of its planarization.

In incremental planarization, the planarization process is split into two stages. First, a large planar subgraph is found within the given graph. Then, the remaining edges that are not already part of this subgraph are added back one at a time, and routed through an embedding of the planar subgraph. When one of these edges crosses an already-embedded edge, the two edges that cross are replaced by two-edge paths, with a new artificial vertex that represents the crossing point placed at the middle of both paths. In some case a third local optimization stage is added to the planarization process, in which edges with many crossings are removed and re-added in an attempt to improve the planarization.

Using incremental planarization for graph drawing is most effective when the first step of the process finds as large a planar graph as possible. Unfortunately, finding the planar subgraph with the maximum possible number of edges (the maximum planar subgraph problem) is NP-hard, and MaxSNP-hard, implying that there probably does not exist a polynomial time algorithm that solves the problem exactly or that approximates it arbitrarily well.

In an n-vertex connected graph, the largest planar subgraph has at most 3n − 6 edges, and any spanning tree forms a planar subgraph with n − 1 edges. Thus, it is easy to approximate the maximum planar subgraph within an approximation ratio of one-third, simply by finding a spanning tree. A better approximation ratio, 9/4, is known, based on a method for finding a large partial 2-tree as a subgraph of the given graph. Alternatively, if it is expected that the planar subgraph will include almost all of the edges of the given graph, leaving only a small number k of non-planar edges for the incremental planarization process, then one can solve the problem exactly by using a fixed-parameter tractable algorithm whose running time is linear in the graph size but non-polynomial in the parameter k. The problem may also be solved exactly by a branch and cut algorithm, with no guarantees on running time, but with good performance in practice. This parameter k is known as the skewness of the graph.

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