Mathematics for Computer Science

11.11. Forests & Trees 353 (b)Verify that for any two verticesx¤yofH 3 , there are 3 paths fromxtoyin H 3 , such that, besid ...

Chapter 11 Simple Graphs354 (c)Explain whyMnis.n1/-edge connected. Problem 11.28. False Claim. If every vertex in a graph has po ...

11.11. Forests & Trees 355 By the principle of induction,P.n/is true for alln 0 , which proves the Claim.  Homework Proble ...

Chapter 11 Simple Graphs356 (d)Explain why all the edges incident to the end ofwmust already be inw. (e)Show that if the end ofw ...

11.11. Forests & Trees 357 graph,G, is connected. This completes the proof of the inductive case, and the Claim follows by s ...

Chapter 11 Simple Graphs358 000 001 010 011 100 101 110 111 Figure 11.27 H 3. Problem 11.35. Procedurecreate-spanning-tree Given ...

11.11. Forests & Trees 359 (b)Prove that any final state of must be a tree on the vertices. (c)For any state,G^0 , letebe th ...

Chapter 11 Simple Graphs360 (b)Conclude Corollary 11.11.11 that if all edges in a weighted graph have distinct weights, then the ...

12 Planar Graphs 12.1 Drawing Graphs in the Plane Suppose there are three dog houses and three human houses, as shown in Fig- ur ...

Chapter 12 Planar Graphs362 Figure 12.1 Three dog houses and and three human houses. Is there a route from each dog house to eac ...

12.2. Definitions of Planar Graphs 363 Figure 12.2 Five quadrapi (4-armed creatures). (a) (b) Figure 12.3 K3;3(a) andK 5 (b). Ca ...

Chapter 12 Planar Graphs364 v u (a) u v (b) Figure 12.4 Planar drawings of (a)K3;3withouthu—vi, and (b)K 5 without hu—vi. Steve ...

12.2. Definitions of Planar Graphs 365 curves cross themselves or other curves, namely, the only points that appear more than on ...

Chapter 12 Planar Graphs366 II I III IV Figure 12.5 A planar drawing with four continuous faces. d c b a II I III IV Figure 12.6 ...

12.2. Definitions of Planar Graphs 367 b e f g c d a Figure 12.7 A planar drawing with abridge. s t u r v x y w Figure 12.8 A pl ...

Chapter 12 Planar Graphs368 12.2.2 A Recursive Definition for Planar Embeddings The association between the continuous faces of ...

12.2. Definitions of Planar Graphs 369 y w a z b x Figure 12.9 The “split a face” case:awxbyzasplits intoawxybaandabyza. z b t v ...

Chapter 12 Planar Graphs370 s t r u u t r s Figure 12.11 Two illustrations of the same embedding. this text, and now that we hav ...

12.3. Euler’s Formula 371 12.3 Euler’s Formula The value of the recursive definition is that it provides a powerful technique fo ...

Chapter 12 Planar Graphs372 graph withvGCvHvertices,eGCeHC 1 edges, andfGCfH 1 faces. Since .vGCvH/.eGCeHC1/C.fGCfH1/ D.vGeGCfG/ ...