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Graph Theory

Graph Theory. Chapter 7 Planar Graphs. 大葉大學 資訊工程系 黃鈴玲 2011.12. Contents. 7.2 Planar Embeddings 7.3 Euler’s Formula and Consequences 7.4 Characterization of Planar Graphs. 7.2 Planar Embedding. Definition.

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Graph Theory

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  1. Graph Theory Chapter 7 Planar Graphs 大葉大學 資訊工程系 黃鈴玲 2011.12

  2. Contents • 7.2 Planar Embeddings • 7.3 Euler’s Formula and Consequences • 7.4 Characterization of Planar Graphs

  3. 7.2 Planar Embedding Definition A graph that can be drawn in the plane without any of its edges intersectingis called a planar graph. A graph that is so drawn in the plane is also said to be embedded (嵌入) in the plane. Applications: (1) circuit layout problems (2) Three house and three utilities(水電瓦斯) problem

  4. (b) (a) Two embeddings of a planar graph

  5. Definition: A planar graph G that is drawn in the plane so that no two edges intersect (that is, G is embedded in the plane) is called a plane graph. (a) planar, not a plane graph (c) anotherplane graph (b) a plane graph

  6. G Note. A given planar graph can give rise to several different plane graph. Definition: Let G be a plane graph. The connected pieces of the plane that remain when the vertices and edges of G are removed are called the regions (or faces)of G. R3: exterior region R1 R2 Ghas 3 regions.

  7. G2 Definition: Every plane graph has exactly one unbounded region, called the exterior region. The vertices and edges of G that are incident with a region R form a subgraph of G called theboundary of R. G2has only 1 region.

  8. G3 v1 v2 v3 v5 v4 v1 v2 v1 v2 v6 v7 v3 v3 v8 v9 v5 v4 v6 v7 v9 Boundary of R1: R1 Boundary of R5: R5 R2 R3 R4 G3has 5 regions.

  9. Observations:(1) Each cycle edge belongs to the boundary of two regions. (2) Each bridge is on the boundary of only one region. (exterior)

  10. Homework Ex: Show that, for every positive integer n, the graph K1,1,n is planar. How many regions result when this graph is embedded in the plane?

  11. 7.3 Euler’ Formula and Consequences Thm 7.14 (Euler’s Formula) If G is a connected plane graph with p vertices, q edges, and r regions, thenp-q + r = 2. pf: (by induction on q) (basis) If q = 0, then G K1; so p = 1, r =1, and p-q + r = 2. (inductive) Assume the result is true for any graph with q = k - 1 edges, where k 1.

  12. Let G be a graph with k edges. Suppose G hasp vertices and r regions. If G is a tree, then G has p vertices, p-1 edges and 1 region.  p-q + r = p – (p-1) + 1 = 2. If G is not a tree, then some edge e of G is on a cycle. Hence G-e is a connected plane graph having p vertices, k-1 edges, and r-1 regions.  p- (k-1) + (r-1) = 2 (by assumption)  p- k + r = 2 #

  13. Homework Ex: Let G be a 5-regular planar graph with 20 vertices. How many regions result when this graph is embedded in the plane?

  14. Definition: A plane graph G is called maximal planar if, for every pair u, v of nonadjacent vertices of G, the graph G+uv is nonplanar. Homework: Draw a maximal planar graph of 6 vertices. How many edges and regions are there in this graph? Thus, in any embedding of a maximal planar graph Gof order at least 3, the boundary of every region of Gis a triangle.

  15. Theorem A: If G is a maximal planar graph with p 3 vertices and q edges, thenq = 3p- 6. Embed the graph G in the plane, resulting in r regions. pf  p-q + r = 2. Since the boundary of every region of G is a triangle, every edge lies on the boundary of two regions.   p-q + 2q/3 = 2.  q = 3p- 6

  16. Corollary: If G is a maximal planar bipartite graph with p 3 vertices and q edges, then q= 2p- 4. The boundary of every region is a 4-cycle. 4r = 2q  p-q + q/2 = 2  q= 2p- 4. pf Corollary: If G is a planar graph with p 3 vertices and q edges, thenq 3p- 6. pf: If G is not maximal planar, we can add edges to G to produce a maximal planar graph. By TheoremA得證.

  17. Corollary 7.18: Every planar graph contains a vertex of degree 5 or less. Let G be a planar graph of p vertices and q edges. pf If deg(v)  6 for every vV(G)   2q  6p 

  18. Two important nonplanar graph K3,3 K5

  19. Observation 7.17 The graphs K5 and K3,3 are nonplanar. (1) K5 has p= 5 vertices and q = 10 edges. q > 3p - 6  K5 is nonplanar. pf (2) Suppose K3,3 is planar, and consider any embedding of K3,3 in the plane. Suppose the embedding has r regions. p-q + r = 2  r = 5 K3,3is bipartite The boundary of every region has 4 edges.  

  20. 7.4 Characterization of Planar Graphs Definition 7.20 Let G be a graph and e={u, v} an edge of G. A subdivisionof eis the replacement of the edge e by a simple path (u0, u1, …, uk), where u0 = uand uk =v are the only vertices of the path in V(G). of G. We say that G’is a subdivision of G if G’ is obtained from G by a sequence of subdivisions of edges in G. 把edge e用一條path取代

  21. H G F Subdivisions of graphs. H is a subdivision of G. F is not a subdivision of G.

  22. Definition Two graphs G’and G’’ are homeomorphicif both G’and G’’ are subdivisions of the same graph G. Observation 7.22 A graph G is planar if, and only if, every graph homeomorphicto G is planar.

  23. Homework • Are the graphs G and H homeomorphic? H G

  24. 1 2 3 1 4 5 6 10 7 4 5 6 2 3 8 9 Thm 7.26: (Kuratowski’s Theorem)A graph G is planar, if and only if, it has no subgraph homeomorphic to either K5 or K3,3. Example 7.30 The Petersen graph is nonplanar. (a) Petersen (b) Subdivision of K3,3

  25. Homework Show that the following graph is nonplanar.

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