To examine whether the given graph $G$ is bipartite, let's proceed step by step.

Definition of a Bipartite Graph:

A graph is bipartite if its set of vertices can be divided into two distinct sets, say $U$ and $V$, such that:

1. Every edge connects a vertex in $U$ to a vertex in $V$, and
2. No edge exists between vertices within the same set.

A useful method for determining whether a graph is bipartite is to attempt to 2-color the graph (i.e., assign two colors to the vertices such that no two adjacent vertices have the same color). If this can be done, the graph is bipartite.

Step 1: Label the Vertices

From the image, the vertices are labeled as $a, b, c, d, e, f, g$.

Step 2: Check for Odd-Length Cycles

A key property of bipartite graphs is that they do not contain odd-length cycles. If an odd-length cycle exists, the graph cannot be bipartite.

Observation:

• Check the cycle $a \rightarrow b \rightarrow d \rightarrow e \rightarrow a$. This is a cycle of length 4, which is even, so it does not disqualify the graph.

• Another cycle: $a \rightarrow f \rightarrow g \rightarrow a$. This is a triangle (3-cycle), which is of odd length.

Since a triangle (3-cycle) exists in the graph, it implies that the graph is not bipartite.

Conclusion:

The graph $G$ is not bipartite because it contains a cycle of odd length (3-cycle).

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What are the main differences between bipartite and non-bipartite graphs?
2. How can you algorithmically check if any graph is bipartite?
3. What are real-life applications of bipartite graphs?
4. Can bipartite graphs have cycles at all, and if so, what kind?
5. How does the absence of odd-length cycles guarantee bipartiteness?

Tip: A quick way to check if a graph is bipartite is to use breadth-first search (BFS) and try coloring the graph with two colors.