4.1.1. (-) Give a proof or a counterexample for each statement below. a) Every graph with connectivity 4 is 2-connected. b) Every 3-connected graph has connectivity 3. c) Every k-connected graph is k-edge-connected. d) Every k-edge-connected graph is k-connected.
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- Give an example of a graph that has all 3 of the following properties. (Note that you need to give a single graph as the answer.) (i) It is connected (ii) It has one articulation point. (iii) The graph needs at least 4 colors for a valid vertex coloring (iv) The graph does not have a 4-clique (that is, a clique of 4 vertices) as a subgraph.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…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
- 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 1Question 1: In graph theory, a graph X is a "complement" of a graph F if which of the following is true? Select one: a. If X is isomorph to F, then X is a complement of F. b. If X has half of the vertices of F (or if F has half of the vertices of X) then X is a complement of F. c. If X has the same vertex set as F, and as its edges ONLY all possible edges NOT contained in F, then X is a complement of F. d. If X is NOT isomorph to F, then X is a complement of F. Question 2: Which statement is NOT true about Merge Sort Algorithm: Select one: a. Merge Sort time complexity for worst case scenarios is: O(n log n) b. Merge Sort is a quadratic sorting algorithm c. Merge Sort key disadvantage is space overhead as compared to Bubble Sort, Selection Sort and Insertion Sort. d. Merge Sort adopts recursive approach3) 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)Draw the following graphs: i. K6 ii. K2,4 iii. C4, that is, the complement of the cycle graph C4 Draw the smallest (i.e. with the least number of vertices and edges) non-trivial (i.e. it has at least one edge) self-complementary graph (GG). Draw all the cubic (i.e. the degree of each vertex equals 3) graphs with at most 8 vertices.5. Icosian Game A century after Euler's discovery (see Problem 4), another famous puzzle-this one invented by the renowned Irish mathematician Sir William Hamilton (1805–1865)-was presented to the world under the name of the Icosian Game. The game's board was a circular wooden board on which the following graph was carved: Find a Hamiltonian circuit-a path that visits all the graph's vertices exactly once before returning to the starting vertex-for this graph.2.4 For each of the following, try to give two different unlabeled graphs with the given properties, or explain why doing so is impossible. (a) Two different trees with the same number of vertices and the same number of edges. A tree is a connected graph with no cycles. (b) Two different connected graphs with 8 vertices all of degree 2. (c) Two different graphs with 5 vertices all of degree 4. (d) Two different graphs with 5 vertices all of degree 3.2. Answer the following questions about the graph given below. B E a) Does an Euler circuit exist? Why or why not? b) Find a good eulerization for the graph, using as few duplicated edges as possible. How many edges must be added? c) Use the eulerization you developed in part b) to determine a best circuit for this graph, beginning at vertex A. List this path by listing successive vertices. 1 3. Consider the complete graph given below. 24 14 46 B, 32 51 28 a) Find the length of the sorted-edges tour for this graph. Use wiggly lines to indicate this tour. b) Find the length of the nearest-neighbor tour starting at vertex A. Use wiggly lines to indicate this tour.Answer True or False to the following claims: a. Let G be a connected graph on 42 vertices and G contains exactly one cycle. The G must contain exactly 42 edges. b. There exists a graph on 42 vertices in which every path has length at most 2. (Recall that a path is not permitted to use the same edge twice in a row.) c. Let G1 be a graph on 5 vertices and 9 edges.Let G2 be a graph on 5 vertices and 9 edges.Then G1 and G2 are necessarily isomorphic.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.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. 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