ExercisesLi Let Ga be the graph af figt.29,la). Find G-U where U={,rie).find GEU] where U=fx*3 *z}Xtel) Find GfF] where Fafe,rtxtlgrEu}-le). find a subgraph H G isomovphic t kz Eg(k) Is there a subyrepifGisomorphictkp+13) what iy the indevi ying si'mple gruphap G?ths what w the intersection of the two subg,vapls you teuned in la)anel (b)2(i) what in the-umion af the tuo subynaphs you founcd in lesandld)?etoCilFigst-2g

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Exercises 1 lt a be the graph af Figrtr29, a). Find G-U where U={x,tzr¥g+¥z}+ es d) Find GfF] where Fafe,rtstlgrtu}- Le). Find a Cil subyraph Hof Gr isomorphic t kg eg k) Is there asubyrephfGisomorphic-to kçr 3) what in the indevi ying-simple gruph ap C2 zhs what w the intersection of the two subyraphs you teuncd in la) and (b) i) what in the umion af the tuo subynaphs yeu founcd in lesandld)? Figstr2g

Exercises 1 lt a be the graph af Figrtr29, a). Find G-U where U={x,tzr¥g+¥z}+ es d) Find GfF] where Fafe,rtstlgrtu}- Le). Find a Cil subyraph Hof Gr isomorphic t kg eg k) Is there asubyrephfGisomorphic-to kçr 3) what in the indevi ying-simple gruph ap C2 zhs what w the intersection of the two subyraphs you teuncd in la) and (b) i) what in the umion af the tuo subynaphs yeu founcd in lesandld)? Figstr2g

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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Let 6 be a simple comected yaph wich vertiles Where n isan number greuker than 2. which one Of the following statements is Of Aruc's theuems Por the Con tapositive graphs A) tea st cnt) Hamil tonian cyue EF ewy rky of G is adsacent to ut 12 vertiles I thn 6 hus B) If 6 hus no lami toniun lycue ten every verter of G j adjacent to at most CntD/2 vertices C) IP G hos no fuleriun Gue then ealh Lertex of 6 is ad jHlent je SS than Cntil2z ertices to D:) Eury urtex Of C has deyrelt at mast nl22 * 6 hus no Hamilronian (41'e
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Q6. Given graph G = (V, E), the graph G'= (V', E') is defined such that (u,v) is in E' if and only if G contains a path with at most two edges between u and v. Describe an efficient algorithm for computing G' from G. Once develop the algorithm assuming G has an adjacency-matrix representation and once develop the algorithm assuming adjacency-list representation for the graph. Analyze the running times of your algorithms.
H.w: show that Let x be a e dge of Connected graph G. The following statements are equivalent. (1) X is X is a bridge of G (2) X is not on any eycle of G ® There exist vertices u and vof G such the ecdgex is on every path goning u and V O There exists a pardition of V into su bsets ų and w such that for any verticesUEU and w the edge x ison every path joining u and w
Q2 a Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is incident to at least one vertex in C. We say that a set of vertices I form an independent set if no edge in G connects two vertices from I. For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of the 20 edges in the graph has at least one endpoint in C, and I = [a, c, i, k] is an independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik. In the example above, notice that each vertex belongs to the vertex cover C or the independent set I. Do you think that this is a coincidence? In the above graph, clearly explain why the maximum size of an independent set is 5. In other words, carefully explain why there does not exist an independent set with 6 or more vertices.
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5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…
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…
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Assignment on Graph A social graph contains all the friendship relations (edges) among a group of n people (vertices). The friendship relationship is symmetric. Two vertices with no edge between them are enemies. Design and implement a class called Graph for a social graph. It should have an adjacency matrix as a data member to represent the graph. The nodes in the graph are numbered from 0..n-1 graph[i][i] - is always true (1) graph[i][j] is true if i and j are friends and false (0) if they are enemies. • In the constructor, initialize all entries to false (0) and all graph[i][i] to true (1) Include the following member functions. o void makeFriends(int i,intj)- make i and j friends int countFriends(int i) - will return the number of friends of i, excluding itself int countEnemies(int i)- will return the number of enemies of i int countCommonFriends(int i,intj) - the number of common friends of i and j int countCommonEnemies(int i,intj) - the number of common enemies of i and j int…
Given the following Graphs: Graph A: В 12c 1 4. E F G- H 4 J K 3 Graph B: Graph B is the undirected version of Graph A. 3.