Suppose G is a simple connected graph with 12 vertices and 16 edges. Suppose 4 of its vertices are degree 1, and 3 of its vertices are degree 2. Prove that G is planar. (Hint: Kuratowski)

prove If G is a plane graph with n vertices, m edges and r regions, then n - m + r = 1 + k(G).

For which of the following does there exist a simple graph G = - (V, E) satisfying the specified conditions? Select one: O A. It has 7 vertices, 10 edges, and more than two components. B. It has 8 vertices, 8 edges, and no cycles. O C. It has 6 vertices, 11 edges, and more than one component. O D. It is connected and has 10 edges, 5 vertices and fewer than 6 cycles. O E. It has 3 components, 20 vertices and 16 edges.

Preview Activity 11.1.1. In this activity we introduce the concept of a double Riemann sum. ………….. a. Review the concept of the Riemann sum from single-variable calculus. Then, explain how we define the definite integral ff(x) dz of a continuous function of a single variable z on an interval [a, b]. Include a sketch of a continuous function on an interval [a,b] with appropriate labeling in order to illustrate your definition. b. In our upcoming study of integral calculus for multivariable functions, we will first extend the idea of the single-variable definite integral to functions of two variables over rectangular domains. To do so, we will need to understand how to partition a rectangle into subrectangles. Let R be rectangular domain R = {(z,y): 0≤x≤6,2 ≤ y ≤ 4} (we can also represent this domain with the notation [0, 6] x [2, 4]), as pictured in Figure 11.1.1. 2 Figure 11.1.1. Rectangular domain R with subrectangles. To form a partition of the full rectangular region, R, we will…