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.

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.

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

Related questions

Q: a search problem on a graph G = (N, E, C), where N represents nodes, E represents edges between…

A: Let's prove whether the heuristic remains admissible and consistent after removing edges from the…

Q: 8. Give a graph with 2n+2 nodes that has 2" shortest paths from node S to node D (specify S and D).

A: Approach: The goal is to determine the shortest path from node S to node D for each query, given a…

Q: ∀p∀q∀r((p≡T ∧q≡F ∧r≡T) →[(p∧¬q)⨁r≡T]) TRUE FALSE because: The contrapositive:

A: Note :We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: How do I do this? We say a graph G = (V, E) has a k-coloring for some positive integer k if we can…

A: Given: How do I do this? We say a graph G = (V, E) has a k-coloring for some positive integer k if…

Q: induced subgraph

A: Given :- In the above question, a statement is mention in the above given question Need to choose…

Q: D I A E H F В G Suppose you run a topological sort on the shown graph starting at vertex B. Use the…

A: It takes as input a directed graph G(V,E) and produces as output set of vertices where each vertex…

Q: 2. Let G = (V, E) be a directed weighted graph with the vertices V = {A, B, C, D, E, F) and…

A: Computer graphics is the field of computer science and technology that involves creating,…

Q: a. Write down the degree of the 16 vertices in the graph below: 14 11 13 12 15 7 10 16

A: A graph can be directed or undirected. A graph in which each vertex is associated with incoming and…

Q: Exercise 4. Let D be a digraph. 1. Suppose that 8+ (D) ≤ 1 and let C be a cycle in D. Show that C is…

A: Directed Cycles:Directed cycles refer to a sequence of vertices in a directed graph where there is…

Q: 5. Vertices: (a, b, c, d, e, f. g} Edges: {{a,b). {a, f). (b, e). (b. g). (c. f). {c. g). (c. d).…

A: vertices {a,b,c,d,e,f,g} edes given draw graph and find adjacent vertices of vertex c find shortest…

Q: Answer True or False to the following claims: a. If G is graph on at most 5 vertices and every…

A: Trees A tree is an undirected, linked, and acyclic graph in graph theory. It is a linked graph…

Q: Given a graph data structure: G = (V,E) where, V = {A, B, C, D, E } E = { (A,B), (A,D), (B,D),…

A: a) Drawing the graph G:``` A / \ B D / / C E ```b) Spanning trees of the graph…

Q: ights are distinct. Is th

A: The statement is true.

Question

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.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 6 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.

Similar questions

We are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…
*Discrete Math In the graph above, let ε = {2, 3}, Let G−ε be the graph that is obtained from G by deleting the edge {2,3}. Let G∗ be the graph that is obtain from G − ε by merging 2 and 3 into a single vertex w. (As in the notes, v is adjacent to w in the new if and only if either {2,v} or {3,v is an edge of G.) (a) Draw G − ε and calculate its chromatic polynomial. (b) Give an example of a vertex coloring that is proper for G − ε, but not for G. (c) Explain, in own words, why no coloring can be proper for G but not proper for G − ε. (d) Draw G∗ and calculate its chromatic polynomial. (e) Verify that, for this example,PG(k) = PG−ε(k) − PG∗ (k).
One can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…
let g be the following dierected graph
Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.
Consider a system of three independent, periodic real-time tasks: T1= (4, 3), T2 = (12,7), and T3 = (8, 5). Construct the schedule for these tasks in the interval [0, 24) with q=1 on twoprocessors with PF. Note that there are many PFair algorithms, while PF is one of them. PF prioritize subtasks by deadlines, and break ties by inspecting future subtask deadlines. Also, if a currently-executing job ties with a newly- released job, then continue executing the currently-executing job; break any other ties by task ID (T1 = highest, T3 = lowest). a. Show all sub jobs’ releases and deadlines for (the first job of) each task. Note that q=1 so each sub job should have execution requirement of 1b. Show the PF schedule.c. Will this task set be schedulable with q=2? Please explain.
Consider eight points on the Cartesian two-dimensional x-y plane. a g C For each pair of vertices u and v, the weight of edge uv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a, h) : V4? + 1? = /17 and dist(a, b) = v2? + 0² = 2. Because many pairs of points have identical distances (e.g. dist(h, c) V5), the above diagram has more than one minimum-weight spanning tree. dist(h, b) = dist(h, f) Determine the total number of minimum-weight spanning trees that exist in the above diagram. Clearly justify your answer.
(V, E) be a connected, undirected graph. Let A = V, B = V, and f(u) = neighbours of u. Select all that are true. Let G = a) f: AB is not a function Ob) f: A B is a function but we cannot always apply the Pigeonhole Principle with this A, B Odf: A B is a function but we cannot always apply the extended Pigeonhole Principle with this A, B d) none of the above
Consider an undirected graph G with 100 nodes. The maximum number of edges to be included in G so that the graph is not connected is
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.
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…
Can you help me solve this exercise? Please note that the greedy approach described in the advice paragraph does not work.