INPUT: A graph G, a non-negative integer k ≥ 0, and a boolean formula F in CNF.  OUTPUT: “Yes” if and only if either G has an independent set of size k or if the boolean formula F is satisfiable.   Prove that this problem is NP-Complete.

• Problem 2 Given undirected graph G=(V,E), with |VM=n, |El=m, and three vertices a,b,c e V. Describe an efficient algorithm that determines whether there is a simple cycle in G that contains all of a, b and c. (No vertex appears more than once within a simple cycle.) Analyze the worst-case time complexity of your algorithm.

Please solve with the computer Question 2:  Draw a simple undirected graph G that has 11 vertices, 7 edges.

Consider eight points on the Cartesian two-dimensional x-y plane. a g C For each pair of vertices u and v, the weight of edge uv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a, h) : V4? + 1? = /17 and dist(a, b) = v2? + 0² = 2. Because many pairs of points have identical distances (e.g. dist(h, c) V5), the above diagram has more than one minimum-weight spanning tree. dist(h, b) = dist(h, f) Determine the total number of minimum-weight spanning trees that exist in the above diagram. Clearly justify your answer.