Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 34.5, Problem 1E
Program Plan Intro
To demonstrate that the sub-graph isomorphism problem is NP-complete.
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determine whether the given pair of graphs is isomorphic. Exhibit an isomorphism or provide a rigorous argument that none exists. Make sure to show what u1 maps to what v, what u2 maps to what v, etc.
Let G be the (3, 4)-complete bipartite graph.
(b) List all non-isomorphic induced subgraphs of G.
Let G be the (3, 4)-complete bipartite graph.(a) List all non-isomorphic subgraphs with 5 vertices of G.(b) List all non-isomorphic induced subgraphs of G.
Chapter 34 Solutions
Introduction to Algorithms
Ch. 34.1 - Prob. 1ECh. 34.1 - Prob. 2ECh. 34.1 - Prob. 3ECh. 34.1 - Prob. 4ECh. 34.1 - Prob. 5ECh. 34.1 - Prob. 6ECh. 34.2 - Prob. 1ECh. 34.2 - Prob. 2ECh. 34.2 - Prob. 3ECh. 34.2 - Prob. 4E
Ch. 34.2 - Prob. 5ECh. 34.2 - Prob. 6ECh. 34.2 - Prob. 7ECh. 34.2 - Prob. 8ECh. 34.2 - Prob. 9ECh. 34.2 - Prob. 10ECh. 34.2 - Prob. 11ECh. 34.3 - Prob. 1ECh. 34.3 - Prob. 2ECh. 34.3 - Prob. 3ECh. 34.3 - Prob. 4ECh. 34.3 - Prob. 5ECh. 34.3 - Prob. 6ECh. 34.3 - Prob. 7ECh. 34.3 - Prob. 8ECh. 34.4 - Prob. 1ECh. 34.4 - Prob. 2ECh. 34.4 - Prob. 3ECh. 34.4 - Prob. 4ECh. 34.4 - Prob. 5ECh. 34.4 - Prob. 6ECh. 34.4 - Prob. 7ECh. 34.5 - Prob. 1ECh. 34.5 - Prob. 2ECh. 34.5 - Prob. 3ECh. 34.5 - Prob. 4ECh. 34.5 - Prob. 5ECh. 34.5 - Prob. 6ECh. 34.5 - Prob. 7ECh. 34.5 - Prob. 8ECh. 34 - Prob. 1PCh. 34 - Prob. 2PCh. 34 - Prob. 3PCh. 34 - Prob. 4P
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- (a) Give the definition of an isomorphism from a graph G to a graph H. (b) Consider the graphs G and H below. Are G and H isomorphic?• If yes, give an isomorphism from G to H. You don’t need to prove that it isan isomorphism.• If no, explain why. If you claim that a graph does not have a certain feature,you must demonstrate that concretely. (c) Consider the degree sequence (1, 2, 4, 4, 5). For each of the following, ifthe answer is yes, draw an example. If the answer is no, explain why. (i) Does there exist a graph with this degree sequence?(ii) Does there exist a simple graph with this degree sequence?arrow_forwardShow that every induced subgraph of a complete graph Kn is also a complete subgraph?arrow_forwardShow that the 3-colorability problem of graphs can be reduced to the 3SAT problem in polynomial time. The 3SAT problem asks, for any given conjunction of disjunctions of at most three Boolean literals, whether the conjunction is satisfiable?arrow_forward
- The third-clique problem is about deciding whether a given graph G = (V, E) has a clique of cardinality at least |V |/3.Show that this problem is NP-complete.arrow_forwardSuppose F, G and H are simple graphs. Suppose f is an isomorphism from F to G and g is an isomorphism from G to H. Which of the following functions h is an isomorphism from F to H?arrow_forwardA clique in an undirected graph is a subgraph wherein every two nodes are connected by an edge. Consider the language: CLIQUE = {G, k : G = (V, E) is an undirected graph containing a clique of size k} Show that 3SAT ≤p CLIQUEarrow_forward
- Let G be a graph with n vertices. The k-coloring problem is to decide whether the vertices of G can be labeled from the set {1, 2, ..., k} such that for every edge (v,w) in the graph, the labels of v and w are different. Is the (n-4)-coloring problem in P or in NP? Give a formal proof for your answer. A 'Yes' or 'No' answer is not sufficient to get a non-zero mark on this question.arrow_forwardDetermine whether the given graphs G and H are isomorphic or not. If they are, provein detail. If they are not isomorphic, explain in detail why they are not.arrow_forwardGive an example of a simple connected graphGfor whichGis planar, has nosubgraph isomorphic toK4, and for whichχ(G) = 4. You must prove with full and completedetails, and a carefully reasoned argument thatχ(G) = 4.arrow_forward
- I really want to learn. Please explain too. For each graph representation, what is the appropriate worst-case time complexity for checking if two distinct vertices are connected. The choices are: O(1), O(V), O(E), or O(V+E) Adjancy Matrix = ____ Edge List = ____ Adjacency List = ____arrow_forwardIf a graph G = (V, E), |V | > 1 has N strongly connected components, and an edge E(u, v) is removed, what are the upper and lower bounds on the number of strongly connected components in the resulting graph? Give an example of each boundary case.arrow_forwardShow that in an undirected graph, classifying an edge .u; / as a tree edge or a back edge according to whether .u; / or .; u/ is encountered first during the depth-first search is equivalent to classifying it according to the ordering of the four types in the classification scheme.arrow_forward
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