Let G be with V .G/ D X [ Y where X D fx1; x2; x3g and Y D fy1; y2; y3; y4; y5g. example of a complete bipartite graph. This particular complete bipartite graph is denoted K3;5. This concept is formally introduced in Definition 52.10. Every vertex in X is adjacent to every vertex in Y , but there are no other edges in G. Please do: a. Find all the maximal independent sets of G. b. Find all the maximum independent sets of G. c. Find all the maximal cliques of G. d. Find all the maximum cliques of G.

Math 2144 Final Review Sections Covered: The ▯nal exam is cumulative. You are responsible for all material covered from Chapters 1 through 5 and Sections 6.1, 6.2, and 6.3. The following review problems cover ONLY the newest content from Chapter 6 and Section 5.8 that you should know for the Final Exam. For your review of material covered from Chapters 1 through 5 you should refer to your previous three exams and the reviews for those exams. Be sure to review carefully the WebAssign homework assignments (accessed under \Past Assignments") and their answer keys. Make sure you know the principles used in solving the problems. ACTUAL FINAL EXAM INSTRUCTIONS: This exam is a closed book exam. You may not use your text, homework, or other aids except for a 3 ▯ 5 notecard. You may use an allowable calculator, TI 83 or 84 to ▯ perform operations on real numbers, ▯ evaluate functions at speci▯c values, and ▯ look at graphs. A TI 89, Nspire, or a calculator with a computer algebra system, any technology with wireless or Internet capability (i.e. laptops, tablets, smart phones or watches), a QWERTY keyboard, or a camera are not allowed. Unless otherwise stated, you must show all of your work including all steps needed to solve each problem and explain your reasoning in order to earn full credit. This means that correct answers using incorrect reasoning may not receive any credit. Reasoning which will earn credit will use material covered in the course. Some short-cuts, such as the using the Fundamental Theorem of Calculus to evaluate a de▯nite