1. Let F be a forest with n vertices and k connected components, with 1 < k < n.(a) Compute ) deg(v) in terms of n and k.vɛV(F)(b) Show that the average degree of a vertex in F is strictly less than 2.(c) Conclude that forests have leaves.

Answer: I have given answer in the handwritten format.I hope tgis will meet your requirement and also hope this will help you lot thanks.

Answered: 1. Let F be a forest with n vertices…

Engineering

Computer Science

1. Let F be a forest with n vertices and k connected components, with 1 < k

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: Provide evidence that the modified version of the choice problem has an NP-complete solution;…

A: The choice problem is a well-known NP-complete problem that involves finding a subset of items with…

Q: Problem 1: Determine whether graphs G and H are planar or not. To show planarity, give a planar…

A: Solution:

Q: Are the following problems in P, NP, co-NP, NP-Hard, NP-complete? Either way, prove it. (a) A kite…

A: Let's analyze the complexities of the given problems:(a) **Max Kite Problem**: - **Complexity…

Q: (V, E), does G contain a path that One-Tenth-Path Problem: Given an undirected graph G = passes…

A: The Subset Sum Problem The Hamiltonian Path Problem The One-Tenth-Path Problem is NP-hard. Given an…

Q: (a) Recall that the path graph on n vertices has vertex set V = {1,.., n} and edge set E = {{i,i+1}…

A: According to the information given:- We have to recall that path graph on n vertices has vertex set…

Q: Given a graph G = (V, E) find whether there exists an independent set of size 8. Is the problem N-P…

A: Dear Student, The answer to your question is given below -

Q: Let n E {3,4,5,6, ...} be fixed. Show that there are exactly two {0,1,2, ...,n – 1} that are…

A: Now consider a DFS through a bridgeless connected graph. Clearly, we will visit each vertex. And…

Q: Prove that if a graph with n vertices and medges is planar and bipartite then m≤2n-4.

A: To prove that if a graph with n vertices and m edges is planar and bipartite, then m ≤ 2n - 4, we…

Q: Problem 3: For each of the two graphs below, determine whether they are planar. Justify your answer:…

A: In graph theory, a planar graph is a graph that can be embedded in the plane, using the kuratowski's…

Q: Problem 4. Let G = (V,E) be an undirected connected graph with maximum edge weight Wmax. Prove that…

A: Introduction: Here we are required to prove that if an edge with weight Wmax appears in some MST of…

Q: Describe a linear-time algorithm for computing the strong connected component containing a given…

A: A directed path between any two vertices defines a subgraph as a strongly connected component (SCC)…

Q: a d b a) Give the vertex set of the G. b) Give the set of edge of G. c) Find the degree of each of…

A: In this question we have been given with the graph and we have to answer for the graph theory…

Q: E F B ប A Determine whether the graph contains an Euler path or an Euler circuit. Select the one…

A: Answer: We need to write the given graph is Euler path or Euler circuit. So we will see in the more…

Q: [Problem 6] Consider an undirected graph G = (V, E), in which each edge has the identical weight.…

A: So one thing you must remember is that a spanning has always has only one incoming edge and has only…

Q: Which of the following problems can be solved in O(n3) time, where n is the number of vertices in…

A: The objective of the question is to identify which of the given problems can be solved in O(n^3)…

Q: ighted graph G with AT MOST 5 vertices which has at leas ees; wo different minimum spanning trees of…

A: Explained

Q: For each of these graphs, find K (G), X(G), and minvev deg (v), and determine which of the two…

Q: The questions in this section refer to this edge-weighted graph: A 7 4 7 12 E

A: Dijkstra's algorithm is a powerful and widely-used algorithm for finding the shortest paths between…

Q: Prove that "If a connected planar simple graph has e edges and v vertices with v ≥ 3 and no circuits…

A: We will prove the statement by induction on the number of vertices, v. Base case: If v = 3, then the…

Q: You are given a graph G, and the problem is to test whether there is a cycle of length at most 1000…

A: To solve this problem, DFS (Initial Search Depth) can be used effectively. By using DFS we obtain…

Q: Consider an undirected random graph of eight vertices. The probability that there is an edge between…

A: Please find the answer below

Q: Provide evidence that the modified version of the choice problem has an NP-complete solution;…

A: The modified version of the choice problem that you are referring to is known as the Maximum…

Question

1. Let F be a forest with n vertices and k connected components, with 1 < k < n.
(a) Compute ) deg(v) in terms of n and k.
vɛV(F)
(b) Show that the average degree of a vertex in F is strictly less than 2.
(c) Conclude that forests have leaves.

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

Step 2

Step 3

Step 4

Trending now

This is a popular solution!

Step by step

Solved in 4 steps with 2 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

Question 4 Given a undirected weighted graph G associate a number which is the weight of the heaviest edge in the tree. Find the spanning tree whose number is minimum among all spanning trees. Explain. (Hint: this involves a simple observation. The question is how to justify this observation. There are many ways to justify. The easiest is probably to recall where "exponentiation" appeared in the HW and use it.) (E,V). With each spanning tree we
From the given information please provide a greedy algorithm and estimate it's time complexity. (will thumbs up) Given: a graph G, find the smallest integer k such that the vertex set V of G contains a set A consisting of k elements satisfying the condition: for each edge of G at least one of its ends is in A. The size of the problem is the number n of vertices in G.
Given: graph G, find the smallest integer k such that the vertex set V of G contains a set A consisting of k elements satisfying the condition: for each edge of G at least one of its ends is in A. The size of the problem is the number n of vertices in G. Please help answer problems 3 & 4 from the given information. 3. Find an instance for which the suggested greedy algorithm gives an erroneous answer. 4. Suggest a (straightforward) algorithm which solves the problem correctly.
(a) Let G be a simple undirected graph with 18 vertices and 53 edges such that the degreesof G are only 3 and 7. Suppose there are a vertices of degree 7 and b vertices ofdegree 3. Find a and b. To receive any credit for this problem you must write completesentences, explain all of your work, and not leave out any details. problem 1, continued(b) Recall that a graph G is said to be k-regular if and only if every vertex in G has degreek. Draw all 3-regular simple graphs with 12 edges (mutually non-isomorphic). Hint:there are six of them. To receive credit for this problem, you should explain, as well aspossible, why all of your graphs are mutually non-isomorphic.
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.
DENSE SUBGRAPH: Given a graph G and two integers a and b, find a set of a vertices of G such that that there are at least b edges between them. Prove that the problem is NP-complete.
Let G be a polyhedral graph with 12 vertices and 30 edges. Prove that the degree of each face of G is three. Show all of your work and clearly explain, using words, your reasoning.
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.
a) write the algorithm b) Briefly explain why the algorithm solves the problem, and the correctness c ) State and briefly explain the running time of your algorithm
Use Euler's inequality (m≤ 3n-6) to prove that each planar graph has at least two vertices of degree at most 5. (Can you do better than 2?)
2. Consider the graph given by the following adjacency list: vertex adjacent vertics 1 2, 3, 4 2 1 3 1, 4 4 1, 3 5 - (a) Draw the corresponding graph. (b) Write down the adjacency matrix A for this graph. Calculate A2. What is the meaning of the entries of matrices A and A2? (c) Starting at vertex 4, use the adjacency matrix algorithm to check for connectivity. Please no plagiarism and also i want clear hand writting. Thanks
Planarity(b) Let G be a simple graph with exactly 11 vertices. Prove that G or its complement G must benon-planar. Hint: The maximum number of edges in a planar graph with n vertices is 3n − 6. for this problem, you must write complete sentences, including all details, showall of your work, and clarify all of your reasoning