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Unformatted text preview: Administration Account forms. I I have some and will hand out after class. I Sign in to class account and register, or we can’t record your grades! CS70: Lecture 5. Outline. 1. Graphs. 2. Tilings. Prove a stronger theorem. 3. Strong Induction. 4. Well Ordering Principle. Reminder: Graphs. Graph: G = ( V , E ) , E ⊆ V × V . Reminder: Graphs. Graph: G = ( V , E ) , E ⊆ V × V . Example: V = { 1 , 2 , 3 , 4 , 5 } E = { } Reminder: Graphs. Graph: G = ( V , E ) , E ⊆ V × V . Example: V = { 1 , 2 , 3 , 4 , 5 } E = { ( 1 , 2 ) , } 1 2 3 4 5 Reminder: Graphs. Graph: G = ( V , E ) , E ⊆ V × V . Example: V = { 1 , 2 , 3 , 4 , 5 } E = { ( 1 , 2 ) , ( 2 , 4 ) , } 1 2 3 4 5 Reminder: Graphs. Graph: G = ( V , E ) , E ⊆ V × V . Example: V = { 1 , 2 , 3 , 4 , 5 } E = { ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 4 ) , } 1 2 3 4 5 Reminder: Graphs. Graph: G = ( V , E ) , E ⊆ V × V . Example: V = { 1 , 2 , 3 , 4 , 5 } E = { ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , } 1 2 3 4 5 Reminder: Graphs. Graph: G = ( V , E ) , E ⊆ V × V . Example: V = { 1 , 2 , 3 , 4 , 5 } E = { ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 1 ) } 1 2 3 4 5 Reminder: Graphs. Graph: G = ( V , E ) , E ⊆ V × V . Example: V = { 1 , 2 , 3 , 4 , 5 } E = { ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 1 ) } 1 2 3 4 5 Reminder: Graphs. Graph: G = ( V , E ) , E ⊆ V × V . Example: V = { 1 , 2 , 3 , 4 , 5 } E = { ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 1 ) } 1 2 3 4 5 For simplicity: No parallel edges! (No loops.) Theorem: Every graph with n nodes and n edges contains a cycle. Theorem: Every graph with n nodes and n edges contains a cycle. A cycle is a sequence of vertices, v 1 ,..., v k where ( v i , v i + 1 ) is an edge and ( v k , v 1 ) is an edge. Theorem: Every graph with n nodes and n edges contains a cycle. A cycle is a sequence of vertices, v 1 ,..., v k where ( v i , v i + 1 ) is an edge and ( v k , v 1 ) is an edge. 1 2 3 4 5 Theorem: Every graph with n nodes and n edges contains a cycle. A cycle is a sequence of vertices, v 1 ,..., v k where ( v i , v i + 1 ) is an edge and ( v k , v 1 ) is an edge. 1 2 3 4 5 Theorem: Every graph with n nodes and n edges contains a cycle. Theorem: Every graph with n nodes and n edges contains a cycle. Proof: By induction. True for n = 1, and 2 trivially (no parallel edges, no loops, so number of edges < n ) Theorem: Every graph with n nodes and n edges contains a cycle. Proof: By induction. True for n = 1, and 2 trivially (no parallel edges, no loops, so number of edges < n ) Base Case: P ( 3 )? 1 2 3 Proof: Induction Step. Proof: Induction Step. P ( n ) = “any n node graph with n edges has a cycle.” Proof: Induction Step....
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This note was uploaded on 02/29/2012 for the course COMPSCI 70 taught by Professor Rau during the Fall '11 term at Berkeley.
 Fall '11
 Rau

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