This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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:...
View
Full
Document
 Spring '11
 Unknown
 Computer Science, Graph Theory, Planar graph, planar graphs, Euler characteristic

Click to edit the document details