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15planar_10

# 15planar_10 - ENGG1007 Foundations of Computer Science...

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1 ENGG1007 Foundations of Computer Science Graphs Graphs Planar Graphs Professor Francis Chin and Dr SM Yiu April 18/19, 2009

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2 ENGG1007 FCS Planar Layouts Planar Layouts Printed Circuit Boards: Can we connect the three pins in chip A to three pins in chip B without crossing the wires? A B Puzzle : Three houses have to connect to three utilities, Electricity, Water and Gas. Houses and Utilities can be treated as vertices and built anywhere . E G W
3 ENGG1007 FCS Planar Graphs Planar Graphs A graph is planar planar if it can be drawn in the plane without any edge crossings. Is K 4 planar? Is Q 3 planar? Is K 3,3 planar?

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4 ENGG1007 FCS Planar Graphs Planar Graphs a b c d h g f e i a b c d h g f e i g a b c d h f e i
5 ENGG1007 FCS North Pole Stereographic Projection Stereographic Projection Points closer to the North Pole will be mapped further away from the sphere. Map every point on a sphere onto a plane (a line extending from the North Pole of the sphere, passing through any point on the sphere and hitting a unique point of the plane). North Pole

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6 ENGG1007 FCS Stereographic Projection Stereographic Projection A graph can be embedded on the surface of a sphere iff it can be embedded in a plane. A planar graph can be embedded in a plane such that any specified region can be made the infinite region (by having the North Pole in that specific region). Any planar graph can be drawn on a plane with all the edges non-crossing and straight .
7 ENGG1007 FCS Formula for Planar Graphs Formula for Planar Graphs A planar drawing of a planar graph divides the plane into a number of regions Example 1: Triangle v = 3, e = 3, r = 2 Example 2: Square v = 4, e = 4, r = 2 Increasing e and v in proportion without increasing r Example 3: Tetrahedron v = 4, e = 6, r = 4 Increasing e and r in proportion without increasing v Example 4: Hypercube Q3 v = 8, e = 12, r = 6 R 3 R 1 R 2 R 4 R 5 R 6 What’s the formula?

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8 ENGG1007 FCS Euler’s Formula for Planar Graphs Euler’s Formula for Planar Graphs For a connected planar simple graph with v vertices, e edges and r regions, r r = A = A e e + B + B v v + C + C Use the above examples to solve A, B and C Use the above examples to solve A, B and C Triangle: v = 3, e = 3, r = 2 2 = 3A + 3B + C Square: v = 4, e
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15planar_10 - ENGG1007 Foundations of Computer Science...

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