## What is a bipartite graph used for?

Bipartite graphs have many applications. They are often used to represent binary relations between two types of objects. A binary relation between two sets A and B is a subset of A × B.

### What are the real world examples of using bipartite graphs?

For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.

#### What is Hall’s theorem in graph theory?

Hall’s marriage theorem says that a graph contains an X-perfect matching if and only if it contains no Hall violators. Assert: M is a matching in G. If M saturates all vertices of X, then return the X-perfect matching M. Let x0 be an unmatched vertex (a vertex in X \ V(M)).

What is true about bipartite graph?

A Bipartite Graph is a graph whose vertices can be divided into two independent sets, U and V such that every edge (u, v) either connects a vertex from U to V or a vertex from V to U. In other words, for every edge (u, v), either u belongs to U and v to V, or u belongs to V and v to U.

Which of the following ways can be used to represent a graph?

Explanation: Adjacency Matrix, Adjacency List and Incidence Matrix are used to represent a graph.

## Are bipartite graphs connected?

bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U and V such that every edge connects a vertex in U to one in V. A bipartite graph doesn’t need to be connected.

### Is every tree a bipartite graph?

Every tree is bipartite. Removing any edge from a tree will separate the tree into 2 connected components.

#### How do you prove a bipartite graph?

The graph is a bipartite graph if:

1. The vertex set of can be partitioned into two disjoint and independent sets and.
2. All the edges from the edge set have one endpoint vertex from the set and another endpoint vertex from the set.

What is Hall’s marriage theorem for bipartite graph?

We will use Hall’s marriage theorem to show that for any m m -regular bipartite graph has a perfect matching. p p vertices from one side of the bipartition. Each vertex has p m.

What is Hall’s theorem?

Hall’s Theorem gives a nice characterization of when such a matching exists. Theorem 1. There is a matching of size Aif and only if every set Sof vertices is connected to at least jSjvertices in B. Proof If such a matching exists, then clearly Smust have at least jSjneighbors just by the edges of the matching.

## Which graph satisfies Hall’s condition and has a perfect matching?

m m -regular bipartite graph has a perfect matching. p p vertices from one side of the bipartition. Each vertex has p m. pm. pm. Each vertex on the other side has degree = p neighbors. Thus, this graph satisfies Hall’s condition and has a perfect matching, as required. The Stable marriage problem is related problem to the marriage problem.