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Unformatted text preview: ENGG1007 oundations of Computer Science Foundations of Computer Science Graphs Graphs Planar Graphs Professor Francis Chin and Dr SM Yiu pril 18/19, 2009 April 18/19, 2009 1 ENGG1007 FCS lanar Layouts lanar Layouts Planar Layouts Planar Layouts • Printed Circuit Boards: • Can we connect the three pins in chip A to three pins chip B without crossing A B in chip B without crossing the wires? Puzzle : Three houses have to connect W to three utilities, Electricity, Water and Gas. Houses and Utilities can be treated as E 2 vertices and built anywhere . G ENGG1007 FCS lanar Graphs lanar Graphs Planar Graphs Planar Graphs A graph is planar planar if it can be drawn in the plane without any edge crossings. ¾ Is K 4 planar? ¾ Is Q 3 planar? ¾ Is K 3,3 planar? 3 ENGG1007 FCS lanar Graphs lanar Graphs Planar Graphs Planar Graphs a b c d e a b c g f g f h i d h e i g a b c 4 d h f e i ENGG1007 FCS tereographic Projection tereographic Projection North Pole Stereographic Projection Stereographic Projection Map every point on a sphere onto a lane line extending from the plane (a line extending from the North Pole of the sphere, passing through any point on the sphere and itting a unique point of the plane) hitting a unique point of the plane). North Pole • Points closer to the North Pole will be mapped further 5 away from the sphere. ENGG1007 FCS tereographic Projection tereographic Projection Stereographic Projection Stereographic Projection A graph can be embedded on the surface of a sphere iff it can be embedded in a plane. planar graph can be embedded in 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 6 noncrossing and straight . ENGG1007 FCS ormula for Planar Graphs ormula for Planar Graphs Formula for Planar Graphs Formula for Planar Graphs • A planar drawing of a planar graph divides the plane into number of gions 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 = 4, = 6, r = 4 , e 6, • Increasing e and r in proportion without increasing v xample 4: ypercube Q3 R 3 R 1 R 2 R 4 R 5 R 6 7 • Example 4: Hypercube Q3 v = 8, e = 12, r = 6 ENGG1007 FCS uler ’s Formula for Planar Graphs uler’s Formula for Planar Graphs 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 = A = A e + B + B v + C + C • Use the above examples to solve A, B and C Use the above examples to solve A, B and C ¾ Triangle: v = 3, e = 3, r = 2 2 = 3A + 3B + C quare: 4 4 2 ¾ Square: v = 4, e = 4, r = 2 2 = 4A + 4B + C implies A + B = 0, C=2 ¾ Tetrahedron:...
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 Spring '11
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 Computer Science

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