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

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Question

Exercise 11.11
An undirected graph G = (V, E) is called bipartite if it contains no odd eycle, 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, i.e., AUB=V and An B = 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.
%3D
The goal of this problem is that, given a connected bipartite graph G = (V, E), to find a partition
of the vertex set into A and B such that all edges are between some verter in A and some vertex
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 vertex v to a. Show that if G is bipartite
then there are no edges between vertices in the same level. (Hint: Use the odd cycle property
of bipartitegraph).
(b) Show that you can output the sets A and B in linear time, ie., O(V|+ JE) time.

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