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Discrete Mathematics with Graph Theory (3rd Edition) 345

Discrete Mathematics with Graph Theory (3rd Edition) 345 -...

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Section 12.4 Exercises 12.4 1. (a) [BB] This is acyclic: F, B, D, H, A, E, C, G is a canonical labeling. (b) This has cycle DH F AED. (c) This is acyclic: C, H, I, F, B, G, D, E, A is a canonical labeling. (d) This digraph has cycle C EF DC. 343 2. indeg Vo = 0 because there are no arcs ViVO with i > O. There are no restrictions on the outdegree of Vo, which can be any integer between 0 and n - 1 (inclusive). 3. [BB] Let 9 be a digraph with n vertices and let A be the adjacency matrix of g. The indegree of vertex i is the sum of the entries in column i. This requires n - 1 two-number additions. Repeating for n vertices involves n(n - 1) = O(n 2 ) additions. 4. (a) We use the algorithm described in Theorem 12.4.3. Given an acyclic digraph with vertex set V = {O, 1,2, ... ,n - I} to find a canonical labeling, Step 1: Let V = {O, 1, ... , n - I} be the vertex set and let t = O. Step 2: While t ~ n - 1 • Let Vt be a vertex in V of indegree O. • Replace V by V '- {Vt} and compute the indegrees of the vertices of (this new) V.
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