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Figure 15.32 A graph
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Figure 15.33 A digraph
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i. Its adjacency matrix j. Its incidence matrix k. Its adjacency list 15.3 15.4 15.5
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[CHAP. 15
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Draw the complete graph on n vertices for n = 2, 3, 4, 5, and 6. Determine whether the graph G1 in Figure 15.34 is either eulerian or hamiltonian. Determine whether the graph G2 in Figure 15.34 is either eulerian or hamiltonian.
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Figure 15.34 Two graphs
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Figure 15.35 shows 12 subgraphs of the graph in Figure 15.6 on page 288. Determine whether each of these is connected, acyclic, and/or spanning.
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Figure 15.35 Twelve graphs
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CHAP. 15]
GRAPHS
Figure 15.35 (continued) Twelve graphs
15.7 15.8 15.9
GRAPHS
[CHAP. 15
Find two nonisomorphic graphs for which all five conditions of Theorem 15.4 on page 289 are true. Describe the adjacency matrix for the complete graph on n vertices. Describe the incidence matrix for the complete graph on n vertices.
15.10 Let G1 be the graph represented by the adjacency list shown in Figure 15.36: a. Draw G1. b. Is G1 a directed graph c. Is G1 strongly connected d. Is G1 weakly connected e. Is G1 acyclic f. Give the adjacency matrix for G1.
Figure 15.36 Adjacency list
15.11 Let G1 be the graph whose adjacency matrix is shown in Figure 15.37: a. Draw G1. b. Is G1 a simple graph c. Is G1 a directed graph d. Is G1 strongly connected e. Is G1 weakly connected f. Is G1 acyclic Figure 15.37 A matrix 15.12 Let G2 be the weighted digraph shown in Figure 15.38: a. Draw the adjacency matrix for this graph. b. Draw the adjacency list for this graph. c. Is this graph connected Justify your answer. d. Is this graph acyclic Justify your answer.
Figure 15.38 A digraph
15.13 A wheel graph on n vertices is a graph of size n +1 consisting of a n-cycle in which each of the n vertices is also adjacent to a single common center vertex. For example, the graph shown in Figure 15.39 is the wheel graph on six vertices. Describe: a. The adjacency matrix of a wheel graph on n vertices b. The incidence matrix of a wheel graph on n vertices c. The adjacency list of a wheel graph on n vertices
Figure 15.39 A graph
CHAP. 15]
GRAPHS
15.14 Let G1 and G2 be the graphs shown in Figure 15.40: a. Determine whether G1 and G2 are isomorphic. Justify your conclusion. b. Either find an euler cycle for G2 or explain why it has none. c. Either find a hamiltonian cycle for G2 or explain why it has none.
Figure 15.40 Two graphs
15.15 Trace Dijkstra s algorithm (Algorithm 15.1 on page 297) on the graph in Figure 15.41, showing the shortest path and its distance from node A to every other node.
Figure 15.41 A weighted graph
Figure 15.42 A weighted graph
15.16 Trace Dijkstra s algorithm on the graph in Figure 15.42, showing the shortest path and its distance from node A to every other node. 15.17 There are four standard algorithms for traversing binary trees: the preorder traversal, the inorder traversal, the postorder traversal. and the level-order traversal. If a binary tree is regarded as a connected acyclic graph, which tree traversal results from a: a. Depth-first search
b. Breadth-first search
15.18 Determine which of the graphs in Figure 15.43 on page 310 are isomorphic. Note that all seven graphs have size 10.
GRAPHS
[CHAP. 15
Figure 15.43 Seven graphs
15.19 For the weighted digraph G1 shown in Figure 15.44 on page 311. a. Draw the adjacency matrix. b. Draw the adjacency list. 15.20 Perform the indicated traversal algorithm on the graph shown in Figure 15.45 on page 311. Give the order of the vertices visited and show the resulting spanning tree: a. Trace the breadth-first search starting at node A. b. Trace the depth-first search of the graph, starting at node A and printing the label of each node when it is visited.
CHAP. 15]
GRAPHS
Figure 15.44 A digraph
Figure 15.45 A graph
Answers to Review Questions
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
15.5 15.6
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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