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Figure 15.44 A digraph
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Figure 15.45 A graph
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15.1 15.2 15.3 15.4 A graph is simple if it has no loops or repeated edges. No: In an undirected graph, an edge cannot be a path because an edge is a set of two elements (i.e., an unordered pair) while a path is a sequence (i.e., an ordered list of vertices). Two vertices are connected if there is a path from one to the other. Two vertices are adjacent if they form an edge. Using only the definition of graph isomorphism, it is easier to prove that two graphs are isomorphic because it only requires finding an isomorphism and verifying that it is one. Proving from the definition that two graphs are not isomorphic would require verifying that every one of the n! one-to-one functions is not an isomorphism. No: The five conditions of are not sufficient for two graphs to be isomorphic. It is possible for all five conditions to be true for two nonisomorphic graphs. (See Problem 15.7.) The reason that the natural definition of a graph prohibits loops is that an edge in a graph is a two-element set, and that requires the two elements to be different. In the natural definition of a digraph, an edge is an ordered pair, and that allows both components to be the same. a. b. c. d. e. f. g. True True True True False True True
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15.5 15.6
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The adjacency matrix is best for a dense graph because it is compact and provides fast direct access. The adjacency list is best for a sparse graph because it allows easy insertion and deletion of edges.
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There must be exactly two 1s in each column of an incidence matrix of a simple graph because each column represents a unique edge of the graph, and each edge is incident upon exactly two distinct vertices.
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Solutions to Problems
15.1 a. b. c. d. e. f. g. h. i. j. k. n = 6. V = {a, b, c, d, e, f}. E = {ab, bc, bd, cd, ce, de, cf, df}. d(a) = 1, d(b) = 3, d(e) = d(f) = 2, d(c) = d(d) = 4. The path abcd has length 3. The path abcfde has length 5. The cycle bcedb has length 4. A spanning tree is shown in Figure 15.46. Its adjacency matrix is shown in Figure 15.47. Its incidence matrix is shown in Figure 15.47. Its adjacency list is shown in Figure 15.47.
Figure 15.46 Spanning tree
Figure 15.47 Adjacency matrix, incidence matrix, and adjacency list
a. b. c. d. e. f. g. h. i. j. k.
n = 6. V = {a, b, c, d, e, f}. E = {ad, ba, bd, cb, cd, ce, cf, de, ec, fe}. id(a) = id(b) = id(c) = id(f) = 1, id(d) = id(e) = 3. od(a) = od(d) = od(e) = od(f) = 1, od(b) = 2, od(c) = 4. The path adec has length 3. The path fecbad has length 5. The cycle adcba has length 4. A spanning tree is shown in Figure 15.48. Its adjacency matrix is shown in Figure 15.49. Its incidence matrix is shown in Figure 15.49.
Figure 15.48 Spanning tree
CHAP. 15]
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l. its adjacency list is shown in Figure 15.49.
Figure 15.49 Adjacency matrix, incidence matrix, and adjacency list
The complete graphs are shown in Figure 15.50:
Figure 15.50 Complete graphs
15.4 15.5 15.6
The graph G1 cannot be eulerian because it has odd degree vertices. But the hamiltonian cycle shown in Figure 15.51 on page 314 verifies that it is hamiltonian. The graph G2 is neither eulerian nor hamiltonian. a. b. c. d. e. f. g. h. i. j. k. l. Disconnected, cyclic, and spanning. Disconnected, acyclic, and spanning. Disconnected, cyclic, and spanning. Disconnected, cyclic, and not spanning. Connected, acyclic, and spanning. Connected, acyclic, and spanning. Connected, cyclic, and spanning. Connected, acyclic, and not spanning. Disconnected, cyclic, and spanning. Connected, cyclic, and not spanning. Disconnected, acyclic, and not spanning. Connected, acyclic, and not spanning.
The two graphs shown in Figure 15.52 on page 314 are not isomorphic because the one on the left has a 4-cycle containing two vertices of degree 2 and the one on the right does not. Yet, all five conditions of Theorem 15.4 on page 289 are satisfied.
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