## 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:

- The vertex set of can be partitioned into two disjoint and independent sets and.
- 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.