apr16 - Illinois Institute of Technology Department of...

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Unformatted text preview: Illinois Institute of Technology Department of Computer Science Lecture 21: April 16 CS 330 Discrete Mathematics Spring Semester, 2008 1 Planar graphs A planar graph is defined as a graph that can be drawn in the plane so that no edges cross. For example the graph on the right is planar, while there is no way to add the edge from ( A, C ) and still have it planar. Note that any graph of less than five nodes must be planar. B A E D C 1.1 Kuratowskis theorem The fully connected graph with five vertices is called K 5 and is isomorphic to the graph on the near right. There is no way that this can be drawn as in the plane with no crossing edges that is, it is a non- planar graph. Another nonplanar graph, called K 3 , 3 , the complete bipartite graph, is on the far right. As it turns out, Kuratowskis theorem states that a graph is nonplanar iff it contains a homeomorphic image of K 3 , 3 or K 5 . The proof of this theorem is beyond the scope of the course. B A E D C A B C D E F 1.2 Eulers formula Notice that when we draw a planar graph, it divides the space (plane) into faces. These faces are regions delimited by edges. On the right are three faces. When edges cross, faces are not well-defined. Also notice that a cycle in a graph determines a face. If there are no cycles, there are no bounded faces (we consider the region outside any cycles to be an unbounded face). This example has five vertices, six edges, and three faces. I II III It turns out that there is a relationship between the number of vertices | V | , the number of edges | E | , and CS 330Spring, 2008 2 Lecture 21: April 16 the number of faces | F | in any planar graph. This result is called Eulers formula and states: For any planar graph: | V | - | E | + | F | = 2 We can test this on out graph and see that 5 - 6 + 3 = 2. This is proved by induction: Base case: | E | = 1. A graph with no cycles has only one face. A graph with only one edge must have two vertices (since this is a connected simple graph), so | V | - |...
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apr16 - Illinois Institute of Technology Department of...

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