11. least one simple circuit of length 6, but G does not have any simple circuit of length 6. a) A classmate claims that these two graphs are not isomorphic because Graph H has at
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- 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.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.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.
- 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 planar1. This question is about type of graphs. a. Construct a graph with exactly 5 nodes such that the graph is strongly connected. b. Construct a graph with exactly 5 nodes such that the graph is weakly connected. c. Construct a graph with exactly 5 nodes such that the graph is completely connected. d. Construct a graph with exactly 5 nodes such that the graph is not connected. e. Construct a graph with exactly 5 nodes such that the graph is not a simple graph.Ex: what are the degrees and what are the neighbourhoods of the vertices in the graphs G and H ? 步 a 'd a
- 5. If the seventh digit of your student ID is an ODD digit, answer Question 5.(a). Otherwise, answer Question 5.(b) [for an EVEN digit]. G 2 12 7 B D (A 1 2 F 3 3 E 10 Fig. 3. Undirected Graph G for Question 5 a) Assume that the vertex A, B, C, D, E, F and G represent seven different places in your neighborhood and the weight of the edges represent the distance in km between these places. Your job is to compute the Minimum Spanning Tree covering the given graph. Explain your computational process in detail. b) Assume that the vertex A, B, C, D, E, F and G represent seven different places in your neighborhood and the weight of the edges represent the distance in km between these places. Your job is to compute the Shortest Path distance from A to other places. Explain your computational process in detail.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…Is it true or false? If it is true, include a (short, but clear) argument why it is true, and if it is false, include a concrete graph which shows that the claim is false.a) If all vertices in a connected graph have even degree, then for whichever two vertices u and v in the graph you choose, there is an Eulerian trail between u and v. b) Given a graph G we construct a new graph H by adding a new vertex v and edges between v and every vertex of G. If G is Hamiltonian, then so is H.c) We know that if a graph has a walk between u and v it also has a path between u and v, for any two vertices u and v. Is it always true that if a graph has a circuit containing u and v it also has a cycle containing u and v? d) The complete bipartite graph K?,?(lowered indicies) is Hamiltonian if and only if m = n ≥ 2.
- 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.The graph below does not have an Euler path and does not have an Euler circuit. F G H a) Find a single edge that if you remove it from the graph, the graph will have an Euler path. b) Suppose you could add edges to the graph of your choosing. What edges can you add to the graph so that the graph will have an Euler circuit? (List the edges you would add in terms of the vertices the edges are incident on.)4. Construct all degree sequences for graphs with four vertices and no isolated vertex. 5. Determine all possible degree sequences for graphs with five vertices containing no isolated vertex and six edges.