The.Algorithm.Design.Manual.Springer-Verlag.1998

Planarity Detection and Embedding
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Planarity Detection and Embedding
Input description: A graph G.
Problem description: Can G be drawn in the plane such that no two edges cross? If so, produce such a
drawing.
Discussion: Planar drawings (or embeddings) make it easy to understand the structure of a given graph
by eliminating crossing edges, which are often confused as additional vertices. Graphs arising in many
applications, such as road networks or printed circuit boards, are naturally planar because they are
defined by surface structures.
Planar graphs have a variety of nice properties, which can be exploited to yield faster algorithms for
many problems on planar graphs. Perhaps the most important property is that every planar graph is
sparse. Euler's formula shows that for planar graph G=(V,E), , so every planar graph
contains a linear number of edges, and further, every planar graph must contain a vertex of degree at
most 5. Since every subgraph of a planar graph is planar, this means that there is always a sequence of
low-degree vertices whose deletion from G eventually leaves the empty graph.
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