Chapter 7 Planar Graphs 大葉大學 資訊工程系 黃鈴玲

 7.2 Planar Embeddings  7.3 Euler’s Formula and Consequences  7.4 Characterization of Planar Graphs 2

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

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

5 (a) planar, not a plane graph 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. (b) a plane graph (c) another plane graph

6 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. Note. A given planar graph can give rise to several different plane graph. R 3 : exterior region G R1R1 R2R2 G has 3 regions.

7 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 the boundary of R. G2G2 G 2 has only 1 region.

8 R1R1 R2R2 R3R3 R4R4 R5R5 G3G3 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 G 3 has 5 regions. Boundary of R 1 : v1v1 v2v2 v3v3 Boundary of R 5 : v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v9v9

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

11 pf: (by induction on q ) Thm 7.14 (Euler’s Formula) If G is a connected plane graph with p vertices, q edges, and r regions, then p  q + r = 2. (basis) If q = 0, then G  K 1 ; 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 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 # Let G be a graph with k edges. Suppose G has p vertices and r regions.

13 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. Thus, in any embedding of a maximal planar graph G of order at least 3, the boundary of every region of G is a triangle. Homework: Draw a maximal planar graph of 6 vertices. How many edges and regions are there in this graph?

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

16 pf 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. Corollary: If G is a planar graph with p  3 vertices and q edges, then q  3p  6. pf: If G is not maximal planar, we can add edges to G to produce a maximal planar graph. By Theorem A 得證. 4r = 2q  p  q + q / 2 = 2  q = 2p  4.

17 pf 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. If deg(v)  6 for every v  V(G) 2q  6p   

18 Two important nonplanar graph K5K5 K 3,3

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

20 Definition 7.20 Let G be a graph and e={u, v} an edge of G. A subdivision of e is the replacement of the edge e by a simple path (u 0, u 1, …, u k ), where u 0 = u and u k =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 Subdivisions of graphs. G HF H is a subdivision of G. F is not a subdivision of G.

22 Definition Two graphs G’ and G’’ are homeomorphic if 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 homeomorphic to G is planar.

23

 Are the graphs G and H homeomorphic? 24 H G

25 Example 7.30 The Petersen graph is nonplanar. (a) Petersen Thm 7.26: (Kuratowski’s Theorem) A graph G is planar, if and only if, it has no subgraph homeomorphic to either K 5 or K 3,3. (b) Subdivision of K 3,

Show that the following graph is nonplanar. 26