Problem 1. In Fig. 1 there is a weighted graph, circles represent vertices, links representedges, and numbers represent edge weights.IN24D3A1B10.31E34HFigure 1: The graph.21T11. Find a shortest path from vertex S to vertex T, i.e., a path of minimum weight betweenS and T.2. Find a minimum subgraph (set of edges) that connects all vertices in the graph andhas the smallest total weight (sum of edge weights).Justify your answers.

Answered: Image /qna-images/answer/6abd2765-a023-474e-9ce0-6abbf0154770.jpg

Answered: Problem 1. In Fig. 1 there is a…

Computer Networking: A Top-Down Approach (7th Edition)

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

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. For each of the following, draw an Eulerian graph that satisfies the conditions, orprove that no such graph exists.a) An even number of vertices, an even number of edges.b) An even number of vertices, an odd number of edges.c) An odd number of vertices, an even number of edges.d) An odd number of vertices, an odd number of edges.
Question 5: Consider the following graph. Find the shortest path from 'a' to all other vertices using Dijkstra's algorithm. Draw graph diagrams to show all intermediate steps. (. 1 4 3 4 a i 3 с 1 2 d 80 g 3
(a) Show that a graph G with at least three vertices is 2-connected if and only if any vertex and any edge of G lie on a common cycle of G: (b) Show that a graph G with at least three vertices is 2-connected if and only if any two edges of G lie on a common cycle
(a) Let G be a simple undirected graph with 18 vertices and 53 edges such that the degreesof G are only 3 and 7. Suppose there are a vertices of degree 7 and b vertices ofdegree 3. Find a and b. To receive any credit for this problem you must write completesentences, explain all of your work, and not leave out any details. problem 1, continued(b) Recall that a graph G is said to be k-regular if and only if every vertex in G has degreek. Draw all 3-regular simple graphs with 12 edges (mutually non-isomorphic). Hint:there are six of them. To receive credit for this problem, you should explain, as well aspossible, why all of your graphs are mutually non-isomorphic.
Consider the directed graph shown below. Which of the following statements are true? b a e d 9 The shortest path from "a" to "g" is of length 3. The shortest path from "a" to "g" is of length 4. The shortest path from "a" to "g" is of length 5. The in-degree of "a" is 1. The in-degree of "a" is 2. O The out-degree of "c" is 1. O The out-degree of "c" is 2. The out-degree of "c" is 3. The graph is acyclic. The ordering "a,b,c,d,e,f,g" is a topological sort for the graph. The ordering "c,b,a,d,f,e,g" is a topological sort for the graph. The graph has 1 strongly connected component. The graph has 2 strongly connected components. The graph has 3 strongly connected components.
4. Draw a Complete Graph, Kn, with n > 4 that has a Hamiltonian Cycle but does not have an Eulerian Cycle. List the degrees of the vertices, draw the Hamiltonian Cycle on the graph and provide justification that there is no Eulerian Cycle.
3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k. Show that the Strongly Independent Set Problem is NP-Complete.
Problem 3 1) Consider that you have a graph with 8 vertices numbered A to H, and the following edges: (A, B), (A, C), (B, C), (B, D), (C, D), (C, E), (D, E), (D, F), (E, G), (F, G), (F, H), (G, H) a) Using depth-first search algorithm, what would be the sequence of visited nodes starting at A (show the content of the stack at some steps). b) Same question if the algorithm used in breadth first search. c) What would be the minimum spanning tree rooted at A if a Depth First Search algorithm is used (refer to question a), show few steps in running the algorithm. d) What would be the minimum spanning tree if a Breadth First Search algorithm is used (refer to question b), show few steps of the algorithm.
18. Show that in a simple graph with at least two vertices there must be two vertices that have the same degree.
(a) Adjacency list and adjacency matrix are two standard ways to represent graphs. Show the space consumption, the cost to count the number of edges in the graph and the cost to check the existence of an edge with adjacency list and adjacency matrix respectively. (b) In Fig. 1 is an undirected graph ?1. Show the sequence of nodes in the breadth-first traversal starting from node ?1, assuming that the neighbor nodes in the adjacency list of each node are in increasing order.
Q.3 (a) Find the degree sequence of the graphs: K6,5, W, (b) Find All values of n such that the graphs: (i) K. (i) C are bipartite (c) Draw the graph with the given incidence matrix 1 100 1011 1 1 100 00 0 01 1001 0 0011 10 0 00 00 0ol Thope the solution is clear and the line is understandable
2. Answer the following questions about the graph given below. B E a) Does an Euler circuit exist? Why or why not? b) Find a good eulerization for the graph, using as few duplicated edges as possible. How many edges must be added? c) Use the eulerization you developed in part b) to determine a best circuit for this graph, beginning at vertex A. List this path by listing successive vertices. 1 3. Consider the complete graph given below. 24 14 46 B, 32 51 28 a) Find the length of the sorted-edges tour for this graph. Use wiggly lines to indicate this tour. b) Find the length of the nearest-neighbor tour starting at vertex A. Use wiggly lines to indicate this tour.

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