Consider the following four graph-theory problems. A. Given a graph G, determine the shortest path from vertex x to vertex y. B. Given a graph G, determine the longest path from vertex x to vertex y. C. Given a graph G, determine whether the graph has a Hamiltonian cycle - i.e., a cycle that passes through every vertex exactly once. D. Given a graph G, determine whether the graph has an Eulerian cycle - i.e., a cycle that passes through every edge exactly once. It turns out that exactly two of these problems can be answered quickly and efficiently, since the running time of the algorithm is polynomial (e.g. O(n³), O(n log n)) rather than exponential (e.g. O(2"), O(n!)) Determine which two problems can be solved in polynomial-time. Your answer will be two letters from the set [A, B, C, D].

Give a sufficient but not necessary condition for a graph to be undirected and without an Eulerian cycle. Please explain your stance.

Problem 7. Determine if a loop-free connected undirected graph with eight vertices, where the degrees of the vertices are 1, 1, 1, 2, 3, 4, 5, and 7, exists. If it exists, draw the graph; otherwise, prove that the graph doesn’t exist.

C. The base function f(x) is reflected in the x-axis, stretched elly by a and translated 3 units to the left and 1 unit down Witte the equation of the function g(x) (4 marks). Given the graph of f(x), graph each of the following clearly labeling each graph: (2 marks each) a. f(-x) f-¹(x) ter of 2. fored b.

Matlab In computer science subject Find the maxima and minima of the given graph 2sint + 0.5t+e

need correct answer 4 this practice question  please

This is a Graph Theory problem.

1. Give all cubic graphs you could find that have order 8, are Hamiltonian, and contain no 3-cycle. Attention: You must draw the 8-cycle first, for each of the graphs you give.

SOLVE STEP BY STEP AND IN DIGITAL FORMAT 3. Prove that every plane graph contains at least two vertices of degree less than or equal to 5.

Problem 2. The two problems below can be solved using graph coloring. For each problem, represent the situation with a graph, say whether you should be coloring vertices or edges and why, and use the coloring to solve the problem. a. Your Quidditch league has 5 teams. You will play a tournament next week in which every team will play every other team once. Each team can play at most one match each day, but there is plenty of time in the day for multiple matches. What is the fewest number of days over which the tournament can take place? b. Ten members of Math Club are driving to a math conference in a neighboring state. However, some of these students have dated in the past, and things are still a little awkward. Each student lists which other students they refuse to share a car with; these conflicts are recorded in the table below. What is the fewest number of cars the club needs to make the trip? Do not worry about running out of seats, just avoid the conflicts. Student: A B C D E F…