Show that the property of being connected is an equivalence relation. That is, for any undirected graph G and any two vertices, v and w, in G, the vertices v and w are related (v ~ w) if and only if there is a path from v to w. Show that v ~ w there is a walk from vertex v to vertex w, then there is a path from vertex v to vertex w. w is an equivalence relation. You can use the fact that if

Show that the property of being connected is an equivalence relation. That is, for any undirected graph G and any two vertices, v and w, in G, the vertices v and w are related (v ~ w) if and only if there is a path from v to w. Show that v ~ w there is a walk from vertex v to vertex w, then there is a path from vertex v to vertex w. w is an equivalence relation. You can use the fact that if

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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Question 1: In graph theory, a graph X is a "complement" of a graph F if which of the following is true? Select one: a. If X is isomorph to F, then X is a complement of F. b. If X has half of the vertices of F (or if F has half of the vertices of X) then X is a complement of F. c. If X has the same vertex set as F, and as its edges ONLY all possible edges NOT contained in F, then X is a complement of F. d. If X is NOT isomorph to F, then X is a complement of F. Question 2: Which statement is NOT true about Merge Sort Algorithm: Select one: a. Merge Sort time complexity for worst case scenarios is: O(n log n) b. Merge Sort is a quadratic sorting algorithm c. Merge Sort key disadvantage is space overhead as compared to Bubble Sort, Selection Sort and Insertion Sort. d. Merge Sort adopts recursive approach
Consider an undirected graph G = (V;E). An independent set is a subset I V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let (G) denote the size of the largest maximal independent set in G. 1) What is (G) if G is a complete graph on n vertices? What if G is a cycle on n vertices?
Answer True or False to the following claims: a. If G is graph on at most 5 vertices and every vertex has degree 2, then G is a cycle. b. Let G be a forest. If we add an edge to G, then G is no longer a forest. c. Let G be a graph, and let u, v, and w be vertices in G. Suppose that a shortest path from u to v in G is of length 3, andsuppose that a shortest path from v to w in G is of length 4. Then a shortest path from u to w in G is of length 7.
For the following simple graphs G_1=(V_1,E_1)G1=(V1,E1) and G_2=(V_2,E_2)G2=(V2,E2) (described by their vertex and edge sets) decide whether they are isomorphic. If they are then prove it, and if they are not give a convincing argument that explains why not: V_1=\{ a,b,c,d,e,f,g,h\}V1={a,b,c,d,e,f,g,h}, E_1=\{ ac,ag,ah,bc,bd,bh,ce,de,df,fg,fh\}E1={ac,ag,ah,bc,bd,bh,ce,de,df,fg,fh}, V_2=\{ 1,2,3,4,5,6,7,8\}V2={1,2,3,4,5,6,7,8}, E_2=\{ 12,15,16,25,27,34,36,37,38,48,67\}E2={12,15,16,25,27,34,36,37,38,48,67}
Consider an undirected graph G = (V;E). An independent set is a subset I V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let (G) denote the size of the largest maximal independent set in G. One way of trying to avoid this dependence on ordering is the use of randomized algorithms. Essentially, by processing the vertices in a random order, you can potentially avoid (with high probability) any particularly bad orderings. So consider the following randomized algorithm for constructing independent sets: @ First, starting with an empty set I, add each vertex of G to I independently with probability p. @ Next, for any edges with both vertices in I, delete one of the two vertices from I (at random). @ Note - in this second step,…
The relations from Exercise 3 are: a. (1,1), (1,3), (2,2), (3,1), (3,3) b. (1,2), (2,2), (3,2) c. (1,1), (1,2), (1,3), (2,1), (2,3), (3,1), (3,2), (3,3) There will be 3 directed graphs. One representing a, then b, then c
Given a vertex set W = {1, 2, 3, 4}, solve for the following questions: (i) Knowing that edges are the same if and only if they have the same endpoint, how many different edges are possible? (ii) Suppose for this question that graphs are different if they have different sets of edges (they do not depend on if the graphs are isomorphic or not). How many simple graphs are there with the vertex set W? (iii) Draw the eleven nonisomorphic simple graphs that have four vertices.
Let G be a connected graph with n vertices and m edges.Which of the following statements are true?(i) G is a tree if and only if n = m+1.(ii) G is a tree if and only if m = n+1.(iii) G is a tree if and only if the addition of any edgeto G will produce a unique cycle.(iv) G is a tree if and only if G contains at least onepath b etween any two vertices.(v) If G is connected, then G is a tree if and only if Gcontains at most one path between any two vertices.(vi) A connected subgraph of a tree is always a tree.
A directed graph G= (V,E) consists of a set of vertices V, and a set of edges E such that each element e in E is an ordered pair (u,v), denoting an edge directed from u to v. In a directed graph, a directed cycle of length three is a triple of vertices (x,y,z) such that each of (x,y) (y,z) and (z,x) is an edge in E. Write a Mapreduce algorithm whose input is a directed graph presented as a list of edges (on a file in HDFS), and whose output is the list of all directed cycles of length three in G. Write the pseudocode for the mappers/reducers methods. Also, assuming that there are M mappers, R reducers, m edges and n vertices -- analyze the (upper-bound of the) communication cost(s).
Let the vertex set V= { 1, 2, 3, 4, 5 } and let the edge set R, a relation on V, be given as follows: R = { (1,1), (1,2), (1,3), (2,1), (2,2), (4,3), (3,5), (4,5), (5,4) } (i) Draw a diagram that represents the relation R graphically (using directed graph). Answer: (ii) Fill the following zero-one matrix representation of the relation R. Answer:
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…
Suppose we have a community of n people. We can create a directed graph from this community as follows: the vertices are people, and there is a directed edge from person A to person B if A would forward a rumor to B. Assume that if there is an edge from A to B, then A will always forward any rumor they hear to B. Notice that this relationship isn’t symmetric: A might gossip to B but not vice versa. Suppose there are m directed edges total, so G = (V, E) is a graph with n vertices and m edges.Define a person P to be influential if for all other people A in the community, there is a directed path from P to A in G. Thus, if you tell a rumor to an influential person P , eventually the rumor will reach everybody. You have a rumor that you’d like to spread, but you don’t have time to tell more than one person, so you’d like to find an influential person to tell the rumor to. In the following questions, assume that G is the directed graph representing the community, and that you have access…