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Unformatted text preview: 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, weve been looking at general properties of graphs and very general classes of relations. Today, well 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 theres 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 isnt. 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. (Ill show one Friday.) Also, crossings are a nuisance in practical design 1 problems for circuits, subways, utility lines. Two crossing connections nor mally means that the edges must be run at different heights. This isnt 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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This note was uploaded on 12/24/2010 for the course CS CS 173 taught by Professor Fleck during the Spring '10 term at University of Illinois, Urbana Champaign.
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