Exercise 11.11 An undirected graph G- (V, E) is called bipartite if it contains no odd cycle, i.e., no cycle has an odd mumber of edges. Note that a graph with no cycles is also bipartite. We are going to show an important property of bipartite graphs its verter set can be partitioned into two disjoint sets A and B, ie., A UB-V and AnB = 6, such that all the edges are between vertices in A and vertices in B- in other words, there are no edges between any two vertices in A and any two vertices in B. The goal of this problem is that, given a connected bipartite graph G= (V, E), to find a partition of the vertez set into A and B such that all edges are between some verter in A and some vertez in B. (a) Do Breadth-First Search (BFS) on G from some starting vertex, say a, and compute the distance (i.e., the level in the BFS tree) of every verteze to a. Show that if G is bipartite then there are no edges betwoen vertices in the same level. (Hint: Use the odd eycde property of bipartitegraph). (b) Show that you can output the sets A and B in linenr time, ie., O(V|+|E]) time.

Exercise 11.11 An undirected graph G- (V, E) is called bipartite if it contains no odd cycle, i.e., no cycle has an odd mumber of edges. Note that a graph with no cycles is also bipartite. We are going to show an important property of bipartite graphs its verter set can be partitioned into two disjoint sets A and B, ie., A UB-V and AnB = 6, such that all the edges are between vertices in A and vertices in B- in other words, there are no edges between any two vertices in A and any two vertices in B. The goal of this problem is that, given a connected bipartite graph G= (V, E), to find a partition of the vertez set into A and B such that all edges are between some verter in A and some vertez in B. (a) Do Breadth-First Search (BFS) on G from some starting vertex, say a, and compute the distance (i.e., the level in the BFS tree) of every verteze to a. Show that if G is bipartite then there are no edges betwoen vertices in the same level. (Hint: Use the odd eycde property of bipartitegraph). (b) Show that you can output the sets A and B in linenr time, ie., O(V|+|E]) time.

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

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…
(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
36. Let G be a simple graph on n vertices and has k components. Then the number m of edges of G satisfies n-k ≤m if G is a null graph. This statement is A. sometimes true B. always true C. never true D. Neither true nor false
*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).
Transcribed Image Text Elon Musk is running the graph construction business. A client has asked for a special graph. A graph is called special if it satisfies the following properties: • It has <105 vertices. It is a simple, undirected, connected 3-regular graph. It has exactly k bridge edges, (k given as input). For a graph G, define f(G) to be the minimum number of edges to be removed from it to make it bipartite. The client doesn't like graphs with a high value of f, so you have to minimize it. If there doesn't exist any special graph, print –1. Otherwise, find a special graph G with the minimum possible value of f(G) and also find a subset of its edges of size f(G) whose removal makes it bipartite. In case there are multiple such graphs, you can output any of those. Elon has assigned this task to you, now vou have to develon a C++ code that takes bridge edges as innut and print all the possible granhs
Please Answer this in Python language: You're given a simple undirected graph G with N vertices and M edges. You have to assign, to each vertex i, a number C; such that 1 ≤ C; ≤ N and Vi‡j, C; ‡ Cj. For any such assignment, we define D; to be the number of neighbours j of i such that C; < C₁. You want to minimise maai[1..N) Di - mini[1..N) Di. Output the minimum possible value of this expression for a valid assignment as described above, and also print the corresponding assignment. Note: The given graph need not be connected. • If there are multiple possible assignments, output anyone. • Since the input is large, prefer using fast input-output methods. Input 1 57 12 13 14 23 24 25 35 Output 2 43251 Q
5. Consider a directed graph G with n nodes. Write a function findUnreachableNode that takes a node and prints all the nodes that are unreachable from the given node. You can use either adjacency list or adjacency matrix to solve this problem. Function Signature: findUnreachableNode (int node) For example: In the following graph, findUnreachableNode (0) will return 4, 6, 7 as they are unreachable from node 0. 1 7 4 Good luck!!!
3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1
3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)
All of the following statements are false. Provide a counterexample for each one. iii. If it can be shown that there is not a proper 3-coloring of a graph G, then χ(G) = 4. iv. If G is a graph with χ(G) ≤ 4 then G is planar
Do some outside research on depth-first traversal as it relates to traversing graphs. Then answer the following questions: a. Suppose you have an arbitrary connected graph G, shown in the image below. Use the vertex A as your starting point. Write out the order in which the algorithm could traverse the graph with a depth-first search, and explain your reasoning (there are multiple correct answers, hence the need for an explanation). b. Use a proof by induction to prove that when a depth-first traversal is performed, every vertex v in your graph G will have been visited at least one time. B D H E A G с I FL
The minimum vertex cover problem is stated as follows: Given an undirected graph G = (V, E) with N vertices and M edges. Find a minimal size subset of vertices X from V such that every edge (u, v) in E is incident on at least one vertex in X. In other words you want to find a minimal subset of vertices that together touch all the edges. For example, the set of vertices X = {a,c} constitutes a minimum vertex cover for the following graph: a---b---c---g d e Formulate the minimum vertex cover problem as a Genetic Algorithm or another form of evolutionary optimization. You may use binary representation, OR any repre- sentation that you think is more appropriate. you should specify: • A fitness function. Give 3 examples of individuals and their fitness values if you are solving the above example. • A set of mutation and/or crossover and/or repair operators. Intelligent operators that are suitable for this particular domain will earn more credit. • A termination criterion for the…