II. Solve the following problems applying the concept of Eulerian and Hamiltonian Graphs. Discuss your procedure in solving each problems. 1. A delivery man is scheduled to transport the orders of items in a certain place. He would like to stroll across all of the streets shown in the figure. (Vertices represent the intersection of streets.) The delivery man does not need to return to the starting place, so the trip can begin from any street intersections. Is it possible for the him to traverse all of the streets exactly once? If yes, help the delivery man design a route for his transactions. A B C D E H

II. Solve the following problems applying the concept of Eulerian and Hamiltonian Graphs. Discuss your procedure in solving each problems. 1. A delivery man is scheduled to transport the orders of items in a certain place. He would like to stroll across all of the streets shown in the figure. (Vertices represent the intersection of streets.) The delivery man does not need to return to the starting place, so the trip can begin from any street intersections. Is it possible for the him to traverse all of the streets exactly once? If yes, help the delivery man design a route for his transactions. A B C D E H

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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a 2. Consider the graph G drawn below. d b e a) Give the vertex set of the G. Give the set of edge of G. b) c) Find the degree of each of the vertices of the graph G. d) Add the degrees of all the vertices of the graph G and compare that sum to the number of edges in G, what do you find? e) Give a path of length 1, of length 2, and of length 3 in G. f) Find the longest path you can in G? h (Remember that you cannot repeat vertices).
Say that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges (u, v), (v, w), (w, t). Show that a graph G with 99 vertices and no path of length three has at most 99 edges.
Draw a simple, connected, weighted graph with 8 vertices and 16 edges, each with unique edge weights. Identify one vertex as a “start” vertex and illustrate a running of Dijkstra’s algorithm on this graph. Problem R-14.23 in the photo
a. Show that if G is a simple graph with n vertices (wheren is a positive integer) and each vertex has degree greater than orequal to (n−1)/2, then the diameter of G is 2 or less. b. If G is a (not necessarily simple) graph with n verticeswhere each vertex has degree greater than or equal to (n−1)/2, is thediameter of G necessarily 2 or less? Either prove that the answer tothis question is "yes" or give a counterexample. Part a has already been answered so I only need help with part b. Part a is included for context
In Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…
a) List all of the paths from 1 to 4 in the graph S with a length of 5. b) The adjacent matrix for the graph R U S
An Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex. NOTE: graphs are in the image attached. Which of the graphs below have Euler paths? Which have Euler circuits? List the degrees of each vertex of the graphs above. Is there a connection between degrees and the existence of Euler paths and circuits? Is it possible for a graph with a degree 1 vertex to have an Euler circuit? If so, draw one. If not, explain why not. What about an Euler path? What if every vertex of the graph has degree 2. Is there an Euler path? An Euler circuit? Draw some graphs. Below is part of a graph. Even though you can only see some of the vertices, can you deduce whether the graph will have an Euler path or circuit? NOTE: graphs is in the image attached.
a. Find an Euler circuit for the graph above. b. If the edge (a-b) is removed from this graph, find an Euler trail for the resulting subgraph. Explain why you are able to find it or why you could not find it for both a and b.
1. Prove that if v1 and v2 are distinct vertices of a graph G = (V,E) and a path exists in G from v1 to v2 , then there is a simple path in G from v1 to v2 .
4. Consider a simple undirected graph with 5 vertices, and let the degrees of these vertices be: 4, 3, 3, 2 and 2. Does such a graph exist? How many edges does such a graph have?
a. Show that K6.7 has a path containing all vertices in the graph. b. Explain why K6,7 is not Hamiltonian.
INSTANCE: A plane graph G = (V, E) (that is a graph embedded in the plane without intersections), and two of its vertices, and t. QUESTION: Does G contain a Hamiltonian path from to t.