Problem 5 Consider a graph G = (V, E) that is a tree and pick any edge e € E. Prove that G' = (V, E - e) is a forest. Explain your solution (premises, axioms and theorems you used) and write all your work to get full credit.
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![Problem 5
Consider a graph G = (V, E) that is a tree and pick any edge e € E. Prove that G' = (V, E – e) is a forest.
Explain your solution (premises, axioms and theorems you used) and write all your work to get full credit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbd250ef0-8825-49ed-a515-c49f8ad0ef73%2F2b8d0e83-ad18-4354-9dee-14132a654fb4%2F3dgefni_processed.png&w=3840&q=75)
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- a 2. Consider the graph G drawn below. d b e a) Give the vertex set of the G. Give the set of edge of G. b) c) Find the degree of each of the vertices of the graph G. d) Add the degrees of all the vertices of the graph G and compare that sum to the number of edges in G, what do you find? e) Give a path of length 1, of length 2, and of length 3 in G. f) Find the longest path you can in G? h (Remember that you cannot repeat vertices).2. (Error-correcting codes x planarity.) Let n be a positive integer. Define the graph H, as follows: • Vertices: the set of all length-n strings over the alphabet {0, 1, 2}. • Edges: draw an edge from string s to string t if the Hamming distance dy(s, t) is 1. (a) Draw H2 (no justification needed.) (b) Prove that for every n 2 3, H, is nonplanar.All of the following statements are false. Provide a counterexample for each one. iii. If it can be shown that there is not a proper 3-coloring of a graph G, then χ(G) = 4. iv. If G is a graph with χ(G) ≤ 4 then G is planar
- . Prove the following.(Note: Provide each an illustration for verification of results)Let H be a spanning subgraph of a graph G.i. If H is Eulerian, then G is Eulerian.ii. If H is Hamiltonian, then G is HamiltonianPlease solve with the computer Question 2: Draw a simple undirected graph G that has 11 vertices, 7 edges.A graph G with 7 edges has that property that all of its vertices have the same degree d. (a) Find all possible values for d. To receive credit for this problem you must write complete sentences, be clear in your reasoning, include all details, and show all of your work. (b) For each such value of d, draw the corresponding graph. (c) TRUE or FALSE (circle one): The complete bipartite graph K20,100 has an Eulerian cycle. Do not justify your answer.
- 3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1Let V= {cities of Metro Manila} and E = {(x; y) | x and y are adjacent cities in Metro Manila.} (a) Draw the graph G defined by G = (V; E). You may use initials to name a vertex representing a city. (b) Apply the Four-Color Theorem to determine the chromatic number of the vertex coloring for G.Draw the following:a. Complete graph with 4 vertices b. Cycle with 3 vertices c. Simple graph with 2 vertices d. simple disconnected graph with 3 vertices e. graph that is not simple. For each of the graphs shown below, determine if it is Hamiltonian and/or Eulerian. If the graph is Hamiltonian, find a Hamilton cycle; if the graph is Eulerian, find an Euler tour.3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…Choose the correct statement. [Choose the most appropriate answer] O None of the option is correct O A graph G has an Euler circuit if and only if G is connected and every vertex of G has an odd degree. O A graph G has an Euler circuit if and only if G is connected and every vertex of G has an even degree. O A graph G has an Euler circuit if and only if G is connected and every vertex of G has an equal degree.SEE MORE QUESTIONSRecommended textbooks for youDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill EducationDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education