Consider a directed graph G = (V, E), and two distinct vertices u, v E V. Recall that a set of U-V paths is non-overlapping if they have no edges in common among them, and a set C of edges disconnects from u if in the graph (V, E-C) there is no path from u to v. Suppose we want to show that for any set of non-overlapping paths P and any disconnecting set C, |P| ≤ |C|. Consider the proof that defines A = P, B = C and f(path q) = qn C, and applies the Pigeonhole Principle to obtain the result. True or False: f is a well-defined function (i.e. it satisfies the 3 properties of a well- defined function). True False

Consider a directed graph G = (V, E), and two distinct vertices u, v E V. Recall that a set of U-V paths is non-overlapping if they have no edges in common among them, and a set C of edges disconnects from u if in the graph (V, E-C) there is no path from u to v. Suppose we want to show that for any set of non-overlapping paths P and any disconnecting set C, |P| ≤ |C|. Consider the proof that defines A = P, B = C and f(path q) = qn C, and applies the Pigeonhole Principle to obtain the result. True or False: f is a well-defined function (i.e. it satisfies the 3 properties of a well- defined function). True False

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

See similar textbooks

Similar questions

Every set of vertices in a graph is biconnected if they are connected by two disjoint paths. In a connected graph, an articulation point is a vertex that would disconnect the graph if it (and its neighbouring edges) were removed. Demonstrate that any graph that lacks articulation points is biconnected. Given two vertices s and t and a path connecting them, use the knowledge that none of the vertices on the path are articulation points to create two disjoint paths connecting s and t.
Define an independent set of a graph G = (V, E) to be a subset S of vertices such that V-S is a vertex cover of G. Is every 2-approximation algorithm for finding a minimum vertex cover also a 2-approximation algorithm for finding a maximum independent set? Justify your answer.
let g be the following dierected graph
Consider a graph with nodes and directed edges and let an edge from node a to node b berepresented by a fact edge (a,b). Define a binary predicate path that is true for nodes c and dif, and only if, there is a path from c to d in the graph.
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).
Hamilton cycle A loop in the connected graph G=(V,E) passes through each vertex in the graph and only once. A Hamiltonian is a path (v,v,.V,.) starting from a certain node v, and looping along the n sides of the graph G. > Except for v,=v, the remaining nodes on the path are different. > (v,V.) EE (0si
Write a function in a directed graph represented by adjacency lists that returns true (1) if an edge exists between two provided vertices u and v and false (0) otherwise.
let us take any standard graph G=(v,e) and let us pretend each edge is the same exact weight. let us think about a minimum spanning tree of the graph G, called T = (V, E' ). under each part a and b illustrate then show that a) s a unique path between u and v in T for all u, v ∈ V . b) tree T is not unique. provide proof
Consider a graph with five nodes labeled A, B, C, D, and E. Let's say we have the following edges with their weights: A to B with weight 3 A to C with weight 1 B to C with weight 2 B to D with weight 1 C to E with weight 4 D to E with weight 2 a. Find the shortest path from A to E using Dijkstra's algorithm b. Use Prim to find the MST c. Use Kruskal to find the MST d. What's the difference between Prim and Kruskal algorithms? Do they always have the same result? Why or why not.
Let G = (V, E) be an undirected graph with at least two distinct vertices a, b ∈ V . Prove that we can assign a direction to each edge e ∈ E as to form a directed acyclic graph G′ where a is a source and b is a sink.
> 4. Let G be an SRG with parameters a, c, and d having n vertices. Show that G is also an SRG and determine its parameters. Moreover, prove the formula a(a-c-1)=(n-a-1)d. Hint: Count the number of edges yz for which y is adjacent to a given vertex a, whereas z is not adjacent to a.
Square graph. The square of a directed graph G = (V,E) is the graph G^2 =(V,E^2)suchthat(u,w)∈E^2 if and only iff or some v∈V both(u,v)∈E and (v, w) ∈ E; that is, there is a path of exactly two edges. (i) Suppose that G is represented by an adjacency list. Describe an efficient algorithm for computing G^2 from G. (ii) Suppose that G is represented by an adjacency matrix. Describe an efficient algorithm for computing G^2 from G.