Graph Planarity. CSC 1300 Discrete Structures Villanova University

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1 Graph Planarity CSC 1300 Discrete Structures Villanova University

2 Major Themes Planar and non-planar graphs The graphs K 5 and K 3,3 Dividing the plane into regions Euler s formula Villanova CSC Dr Papalaskari 2

3 Three house/ three ullity puzzle Is it possible to join three houses to three separate ulliles so that none of the conneclons cross? gas water electric Villanova CSC Dr Papalaskari 3

4 a c Planar graphs f b d A graph is planar if it can be drawn in the plane without any edges crossing e Villanova CSC Dr Papalaskari 4

5 More examples Geogebra files available at hvp:// Source: Discrete MathemaLcs with Ducks by Sara-Marie Belcastro, 2012, CRC Press, Fig Villanova CSC Dr Papalaskari 5

6 More examples Geogebra files available at hvp:// Source: Discrete MathemaLcs with Ducks by Sara-Marie Belcastro, 2012, CRC Press, Fig Villanova CSC Dr Papalaskari 6

7 K 3,3 is not planar Consider region R formed by verlces 1, 4, 2, 5. Case 1: vertex 3 inside R Case 2: vertex 3 outside R Where can vertex 6 go? R 3? 5 2? Villanova CSC Dr Papalaskari 7

8 K 5 is not planar Prove using a similar argument based on regions. Villanova CSC Dr Papalaskari 8

9 Kuratowski s Theorem Are there non-planar graphs other than K 3,3 and K 5? What makes a graph non-planar? If a graph has K3,3 or K5 as a subgraph If a graph has a subgraph that can be obtained by adding intermediate verlces to edges of either K 3,3 or K 5 (such a graph is called a subdivision) any others? graph homeomorphic to K 5 Kuratowski s Theorem: A graph is nonplanar if and only if it contains a subgraph that is a subdivision of either K3,3 or K5. Villanova CSC Dr Papalaskari 10

10 Previous example non-planar Here s a problem: contains K 3,3 subgraph! Source: Discrete MathemaLcs with Ducks by Sara-Marie Belcastro, 2012, CRC Press, Fig Villanova CSC Dr Papalaskari 11

11 Regions of the plane Planar graphs divide the plane into regions Villanova CSC Dr Papalaskari 12

12 Redraw, count regions: Villanova CSC Dr Papalaskari 13

13 Example: RelaLng number of regions to verlces and edges Consider the following problem involving rice fields separated by walls. How many edges (walls) can we remove before all the rice fields are flooded? What type of graph are we leh with aher all these walls are removed? Number of regions: r= Number of verlces: n = Number of edges: e = edges removed in flood: e 1 = edges remaining: e 2 = Source: Discrete MathemaLcs with Ducks by Sara-Marie Belcastro, 2012, CRC Press, p317. Villanova CSC Dr Papalaskari 14

14 Euler s IdenLty for planar graphs Theorem: Let G be a connected planar graph with n verlces, e edges and r regions. Then: n e + r = 2 Villanova CSC Dr Papalaskari 15

15 Example Let G be a connected 3-regular graph of order 8. How many regions does G have? Villanova CSC Dr Papalaskari 16

16 A corollary of Euler s IdenLty helps rule out planarity for some graphs Corollary: If G is connected planar graph, then e 3n - 6 IntuiLon: graphs with lots of edges cannot be planar! Example: What can we say about K 5? K 3,3? Villanova CSC Dr Papalaskari 17

17 Previous examples e = n = e > 3n 6? Planar? Villanova CSC Dr Papalaskari 18

18 Previous examples e = n = e > 3n 6? e = n = e > 3n 6? Planar? Planar? Villanova CSC Dr Papalaskari 19

Major Themes. Graph Coloring. Vertex Colorings. Dr Papalaskari 1

Major Themes. Graph Coloring. Vertex Colorings. Dr Papalaskari 1 Major Themes Graph Coloring CSC 1300 Discrete Structures Villanova University Vertex coloring ChromaFc number χ(g) Map coloring Greedy coloring algorithm ApplicaFons Villanova CSC 1300 - Dr Papalaskari