Consider the above rooted tree. 1. Which vertex is the root? 2. Which vertices are internal? 3. Which vertices are leaves? 4. Which vertices are children of j? 5. Which vertex is the parent of h? 6. Which vertices are siblings of o? 7. Which vertices are ancestors of m? 8. Which vertices are descendants of b?

Use a tree to model the parent-child relationships in the following family. Paloma has three children: Sherry, Laura, and Kerry. Sherry has one child, Donna. Laura has one child, Betty. Kerry has one child, Hopalong. Use vertices to model the people and edges to represent the parent-child relationships. Label the vertices with the person's initial. Choose the graph that represents the parent-child relationships. O A. O C. K L 14 L K Au P H B D S M B. P HBD S Au H B D S O D. K L P • C H B D S

50. Use the Breadth-First Spanning Tree Search Algorithm to find a spanning tree of K,. Start from vertex h. Remember, we start with vertex h being labeled 0. b K6 a. Vertices that should be labeled as step 1: b. Vertices that should be labeled as step 2: c. Vertices that should be labeled as step 3: d. Edges that need to be removed to get a spanning tree:

Consider the following graph. Start from vertex x, use the Depth First Search (DFS) algorithm (described in the textbook) to compute the DFS tree(s). Show intermediate results similar to the textbook example and show the final DFS tree(s). It is assumed that in the adjacency list representation, all vertices are stored according to their alphabetical order. In other words, a vertex’s neighbors should be explored by their relative alphabetical order.

a. Draw a graph with 5 vertices and 5 edges. Can such a graph be a tree? Carefully explain your answer.

4. Consider the graph G, shown below, which is simply a copy of K5. 02 V3 5 V1 24 V5 How many distinct spanning trees does G have? (Hint: Break up your search by the isomorphism type of the tree, as discovered on the previous page. So for example, start by counting the paths of length 5 in G. Then proceed to the next type of tree with 5 vertices. The total number of trees is 125, but please use this answer only to check that your solution is complete!)

Find the articulation vertices in the following graph starting at root node #2. What property do you think the root should have so that it can be checked, just like other nodes, for being an articulation vertex? 2 8 7 6 5

Draw the minimum spanning tree for the following graph. Use induction to prove your result is a minimum spanning tree.