f2006lecNotesWeek12 - 33 33.1 Planar Graphs Introduction to...

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33 Planar Graphs 33.1 Introduction to planar graphs defnition a graph is planar iF it can be drawn in the plane without any intersecting edges. concept oF a Face in a planar graph a face is a region oF the plane bounded by the edges and vertices oF a circuit this includes the part oF the plane bounded by the exterior edges and vertices oF the graph any two points in a Face can be joined without crossing an edge oF the graph 33.2 Euler’s formula in a connected planar graph G =( V,E ) let v = | V | let e = | E | let f be number oF Faces then Euler’s Formula states that v - e + f =2 examples inFormal prooF start with one vertex (hence no edges, one Face) v - e + f =1 - 0+1=2 then add edges, one at a time, always maintaining a connected, planar graph iF edge joins a new vertex to an existing vertex e e +1 v v f unchanged v - e + f is unchanged iF edge joins two existing vertices f f e e v unchanged v - e + f is unchanged application to polyhedra pyramids cubes 33.3 Determining Planarity K 5 is not planar demonstrate that an attempt to construct a planar version oF K 5 must Fail start with three vertices graph is planar with two Faces: f 1 and f 2 •• v 1 v 2 v 3 f 1 f 2 ....................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . now add a Fourth vertex v 4 and edges to frst three vertices 72
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•• v 1 v 2 v 3 v 4 f 11 f 12 f 13 f 2 ....................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ........ ... . . ... .. ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . ..... ..... ..... . .... . ..... . . or v 1 v 2 v 3 v 4 f 1 f 21 f 22 f 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ... . .. .. .... .. . .... . .... ... . ... . . ... . ..... . . . . . .... . ... ...... ....................... in either case, we have four faces, each bounded by edges incident on three vertices if we now try to place v 5 in any face, it can only be joined to the three vertices in that face the fourth vertex cannot be reached (if planarity is to be retained) can take a more algebraic approach use Euler’s formula to show that in a connected simple planar graph, e 3 v - 6 but, for K 5 , e
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f2006lecNotesWeek12 - 33 33.1 Planar Graphs Introduction to...

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