In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. Minimize C = 10 x 1 + 15 x 2 subject to − 4 x 1 + x 2 ≥ 12 12 x 1 − 3 x 2 ≥ 0 x 1 , x 2 ≥ 0
In Problems 13-20, (A) Form the dual problem. (B) Find the solution to the original problem by applying the simplex method to the dual problem. Minimize C = 10 x 1 + 15 x 2 subject to − 4 x 1 + x 2 ≥ 12 12 x 1 − 3 x 2 ≥ 0 x 1 , x 2 ≥ 0
Solution Summary: The author explains how to determine the dual of the minimization problem using the coefficients in the problem constraints and the objective function.
Give an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.
3. [10 marks]
Let Go (Vo, Eo) and G₁
=
(V1, E1) be two graphs that
⚫ have at least 2 vertices each,
⚫are disjoint (i.e., Von V₁ = 0),
⚫ and are both Eulerian.
Consider connecting Go and G₁ by adding a set of new edges F, where each new edge
has one end in Vo and the other end in V₁.
(a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so
that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian?
(b) If so, what is the size of the smallest possible F?
Prove that your answers are correct.
Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.
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