Is It Possible in a Bipartite Graph for a Vertex to be Unconnected to Another Vertex of an Independent Set?

In a bipartite graph, the vertex set is divided into two disjoint independent sets, typically referred to as (U) and (V). By definition, there are no edges between vertices within the same independent set. However, each vertex in one independent set can be connected to one or more vertices in the other independent set. It is indeed possible for a vertex in one independent set to not be connected to any vertex in the other independent set. This situation occurs when there are no edges incident to that vertex, leading to a situation where it is isolated with respect to the other set.

Explanation and Example

Consider a bipartite graph with the following sets:
- (U {u_1, u_2})
- (V {v_1, v_2, v_3})

If the edges are defined as:
- (E {u_1v_1, u_1v_2})

In this case:
- (u_2) is not connected to any vertex in (V) i.e. (v_1, v_2, v_3).
- (v_3) is isolated with respect to (U).

Thus, a vertex in one independent set can indeed be unconnected to any vertex in the other independent set.

Can a Bipartite Graph Exist Without All Vertices Being Connected?

Yes, it is perfectly possible in a bipartite graph that any vertex does not need to be connected with another vertex of the independent set. A 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 does not need to be connected. It is entirely fine to have (U) and (V) of any size, possibly even empty. We can also have any number of edges, possibly even zero.

Example: A Non-Connected Bipartite Graph

Here's an example of a bipartite graph with (U) and (V) indicated by vertex colors. The graph drawn in the question is a complete bipartite graph (K_{1,2}), which we can highlight as shown below:

[Graph Image: A bipartite graph with U and V shown in two different colors, and edges connecting all vertices in U to all vertices in V]

In (K_{1,2}), vertex (u_1) is connected to both (v_1) and (v_2), while (u_2) and (v_3) are not connected to any vertex in the opposite set, demonstrating the possibility of unconnected vertices in a bipartite graph.