7.4 K urato wskiÕ s Theorem 7.5 Alg ebraic T ests fo r ...cs4203/files/GT-Lec9.pdf · Chapter 7 Planar ity B: Lec 9 7.4 K urato wskiÕ s Theorem 7.5 Alg ebraic T ests fo r Planarity

Chapter 7Planarity B: Lec 9

7.4 Kuratowski’s Theorem7.5 Algebraic Tests for Planarity

1 c 2010 by Jonathan L. Gross

2 Chapter 7 Planarity B: Lec 9

7.4 KURATOWSKI’S THEOREM

c 2010 by Jonathan L. Gross

Section 7.4 Kuratowski’s Theorem 3



Step 1: Min Counterex Is Simple & 3-Conn



Step 2: Three Overlapping Appendages

Fig 7.4.3



Fig 7.4.3





Step 3: Cycle+Appendages Configuration

Fig 7.4.5



Fig 7.4.6



Fig 7.4.7



Fig 7.4.8



Fig 7.4.9



Finding K or K in Small Nonplanar Graphs

Fig 7.4.10



Fig 7.4.11

Fig 7.4.12



Fig 7.4.13



7.5 ALGEBRAIC PLANARITY TESTS

Fig 7.5.1


Section 7.5 Algebraic Planarity Tests 17



Fig 7.5.2



Fig 7.5.3



Face-Size Equation

Fig 7.5.4



Edge-Face Inequality



Euler Polyhedral Equation



Fig 7.5.6





Fig 7.5.7



Poincaré Duality

Fig 7.5.8



Fig 7.5.9



Fig 7.5.10



Fig 7.5.11



Algebraic Proofs of Nonplanarity





Nonplanarity Condition for Bipartite Graphs



Nonplanar Subgraphs



Fig 7.5.12





Fig 7.5.14


Section 7.8 Supplementary Exercises 37

7.8 SUPPLEMENTARY EXERCISES


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7.4 K urato wskiÕ s Theorem 7.5 Alg ebraic T ests fo r ...cs4203/files/GT-Lec9.pdf · Chapter 7 Planar ity B: Lec 9 7.4 K urato wskiÕ s Theorem 7.5 Alg ebraic T ests fo r Planarity