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lect_37 - Planar Graphs I Margaret M Fleck 28 April 2010...

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Planar Graphs I Margaret M. Fleck 28 April 2010 This is a half-lecture 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 self-loops or multi-edges. 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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