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4. Find the minimum spanning tree. C B 1 2 4 , we have U = {41,11,141,us} - 5. Convert the graph in #4 into a tree using BFS and DFS starting with vertex A. 6. Determine the stable matchings resulting from the Proposal Algorithm run when men proposing and with women proposing, given the preference lists below. 1 D BCA E A 4 2 5 3 1 2 ECB A D 2 1 4 3 5 3 BE ADC C 3 5 4 4 E BDC A D 1 3 2 5 5 D A B C E E 2 5 1 3 4 7. For bipartite graph G=(U,V,E) V = {v}, V2, V5, va} and E = {(1,,vi),(u, v),(4,,v.), (uz,v;), (uz, v3), (uz,va),(uz,vi),(uz, vs),(un, v;), (u, v}} a. Draw G. b. Does G have a maximal matching of size 1? Size 2? Size 3? For each yes, draw an auxiliary graph depicting the matching and provide an alternating path P to find a larger matching. 8. Find the minimum-weight transversal in the matrix below. 1 2 3 5 A 4 5 8 10 11 7 6 7 4 8 5 12 9 6 10 7 E 4 5 7 9 8 9. At a school ice cream party there are five dixie cups of ice cream that remain to be served. Each cup has a different favor: vanilla, chocolate, cherry, rocky road, and mint and chip. There are five children who have yet to be served: A, B, C, D and E. The ice-cream preferences of the children are shown below. Child Vanilla Chocolate Cherry Rocky Road Mint & Chip A X B C X 2/3 4 B 5 D 6 6 13 D X In a rush to get their ice cream, A grabbed the cherry, C the chocolate, D the mint and chip, and E the rocky road. This left B with a (vanilla) flavor that he does not like, and which he refused to eat. Show how matching algorithm can be used to increase the current matching. Make a matching that assigned their preferred ice cream to each child. 10. Determine whether the graph has 1 factor