1. Construct a simple graph that is a forest with vertices M, N, O, P, Q, R such that the degree of O is 2 and there are 2 components. What is the edge set?

One can model a maze by having a vertex for a starting point, a finishing point, dead ends, and all the points in the maze where more than one path can be taken, and then connecting the vertices according to the paths in the maze. a. Construct such a graph for the following maze. 鷗 b. Which traversal DFS or BFS- would you use if you found yourself in a maze and why?

Given a graph that is a tree (connected and acyclic). (1) Pick any vertex v. (II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance. (III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are true a. p is the longest path in the graph b. p is the shortest path in the graph c. p can be calculated in time linear in the number of edges/vertices a,c a,b a,b,c b.c

Construct the simple graph of 4 vertices, where degree(v1)=3,degree(v2)=2,degree(v3)=2,degree(v4)=1 What is the total number of edges of the complete simple graph of 4 vertices Write a formula to find the total number of edges in a complete simple graph based on n where n is the total number of vertices. How many adjacent vertices to a vertix of degree 3

2. Consider the graph G2(V2, E2) below. 2a. Find the MST of this graph with Kruskal’s algorithm. Draw the MST, and show the table [edge] [w(u, v)] [mark]. 2b. Find the MST of this graph with Prim’s algorithm starting at vertex 'a'. Draw the MST and list the vertices in order you added them to the MST. 2c. Would you get a different MST if you repeat 2b starting at vertex 'e'? Why or why not?

Weighted Graph Applications Demonstration Java Data Structures. Figure 29.23 illustrates a weighted graph with 6 vertices and 8 weighted edges. Simply provide: Minimal Spanning Tree as an illustration or a textual list of edges (in our standard vertex order). Single-Source Shortest Path route from vertex 0 to the other 5 (described as one path/route for each). draw the two solutions and attach the illustration or describe them in text (a list of edges for the one and the vertex to vertex path the other). You can therefore attach proper content files with dot txt, png, jpg or jpeg extensions  Be sure the final trees or path lists are clearly visible in your solution.  You don't need to show the solution development or progress, just the result.

1. This question is about type of graphs. a. Construct a graph with exactly 5 nodes such that the graph is strongly connected. b. Construct a graph with exactly 5 nodes such that the graph is weakly connected. c. Construct a graph with exactly 5 nodes such that the graph is completely connected. d. Construct a graph with exactly 5 nodes such that the graph is not connected. e. Construct a graph with exactly 5 nodes such that the graph is not a simple graph.

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

an undirected graph is called complete if every shares an edge with every other vertex.

Give an example of a graph that has all 3 of the following properties. (Note that you need to give a single graph as the answer.) (i)  It is connected (ii)  It has one articulation point. (iii)  The graph needs at least 4 colors for a valid vertex coloring (iv)  The graph does not have a 4-clique (that is, a clique of 4 vertices) as a subgraph.