Planar Graphs I
Margaret M. Fleck
28 April 2010
This is a halflecture due to the third quiz.
This lecture surveys facts about graphs that can be drawn in the plane
without any edges crossing (first half of section 9.7 of Rosen).
1
Planar graphs
So far, we’ve been looking at general properties of graphs and very general
classes of relations.
Today, we’ll concentrate on a limited class of graph:
simple undirected connected graphs. Recall that a simple graph contains no
selfloops or multiedges. and connected means that there’s a path between
any two vertices. And we assume (without ever saying this explicitly) that
all graphs are finite.
Which of these graphs are “planar” i.e. can be drawn in the plane without
any edges crossing (i.e. not at a vertex)?
Examples:
K
4
is planar, cube (
Q
3
) is planar,
K
3
,
3
isn’t. See pictures in
Rosen p. 658.
Notice that some pictures of a planar graph may have crossing edges.
What makes it planar is that you can draw at least one picture of the graph
with no crossings.
Why should we care? Connected to a variety of neat results in mathemat
ics. (I’ll show one Friday.) Also, crossings are a nuisance in practical design
1
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problems for circuits, subways, utility lines. Two crossing connections nor
mally means that the edges must be run at different heights. This isn’t a big
issue for electrical wires, but it creates extra expense for some types of lines
e.g.
burying one subway tunnel under another (and therefore deeper than
you would ordinarily need). Circuits, in particular, are easier to manufacture
if their connections live in fewer layers.
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 Spring '10
 fleck
 Graph Theory, Planar graph

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