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Doc6 - 5| Fer rt 3 2 let G =(V E he the leep-free...

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Unformatted text preview: 5|. Fer rt 3 2, let G = (V. E ) he the leep-free undirected graph, where V is the set ef binary n-tuples [ef 0’s and 1’s} and E = {{u, w}|v. tn E V and 1:, ts differ in [exactly] twe pesitiens}. Find MG). '1". SEVEN tewns rt, h, c, d, c, If, and g are connected by a sys- tem ef highways as fellews: (1) 1-22 gees frent a te c, passing threugh b; (2}1-33 gees frern c re d and then passes threttgh b as itcentinues te f ; [3} [-44 gees fretn d threugh s te n; [4] I-55 gees item f In is, passing threugh g; and [5) 1—65 gees from g to d . ll]. Give an example ef a cennected graph G where rernevittg an}? edge ef G results in a discnnnected graph. 12. a] If G = (V. E) is an undirected graph 1with |V| = v. |E| = e, and ne leeps, pres-e that 2.9 1: u1 — 1.1. h) State the eert'espettding, inequality»r fer the case when G is directed. 3. a] Haw man}r spanning subgraphs are there fer the graph G in Fig. ll.2?(a}? h} Haw many ednneeted spanning suhgraphs are [here it] Part (a)? e) Haw man}r ef the spanning subgraphs in part (a) have vertex {4 as an isolated vertex?I 15. Find all [leap-free} nenise-merphie undireeted graphs with fear vertiees. Haw many of these graphs are connected“? 12. a} Let G be an undirected graph with n vertiees. If G is isn— marphie ta its awn complement 6+ haw manyr edges must G hawe‘.l (Sueh a graph is ealled seifieempfemeatary.) h} Find an example of a self-eemplementaiy graph an four 1l'ertiees and one em five vertiees. e} If G is a self—eemplementary graph en a vertiees, where a e: Lpreve thatn = 4!: are = M: + 1, far semefr E T“. ...
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