To analyze which vertices have a path between them, we need to examine the graph structure based on its connectivity. Here are the steps to identify the paths between vertices:

Step 1: Identify Connected Components

The graph you provided appears to be composed of two connected components:

1. Left Component: Consists of vertices $A, B, C, D, E, F, G, H, I, J, K$.
2. Right Component: Consists of vertices $L, M, N, O, P, Q, R, S, T$.

• Vertices within each component have paths between them because they are directly or indirectly connected by edges.
• There is an edge connecting the left component (vertex $G$) and the right component (vertex $N$), so all vertices in the graph can reach each other.

Step 2: Paths between Vertices

• From left to right: Any vertex from the left component can reach any vertex in the right component via the connection $G \leftrightarrow N$.
• From right to left: Similarly, any vertex from the right component can reach any vertex in the left component via the same connection.

Conclusion:

• All vertices in the graph are connected either directly or through other vertices, meaning there is a path between every pair of vertices.

Would you like more details on specific paths between vertices, or help with analyzing the structure further?

Related Questions:

1. What is the shortest path between vertex $A$ and vertex $Q$?
2. How can we compute the degree of each vertex in the graph?
3. How many edges are there in the graph in total?
4. What is the diameter of this graph (longest shortest path)?
5. Can we find the minimum spanning tree of this graph?

Tip: To find paths efficiently in graphs, algorithms like breadth-first search (BFS) are useful for unweighted graphs like this one.