This problem asks whether it's possible to traverse a city map (modeled as a graph) by crossing each bridge exactly once. This is a classical Eulerian path or Eulerian circuit problem.

Key points:

• The vertices (A, B, C, D, E) represent the land masses.
• The edges (the red lines) represent the bridges.
• An Eulerian path exists if and only if the graph has exactly 0 or 2 vertices with odd degrees (the number of edges connected to each vertex).
• If there are 0 odd-degree vertices, the Eulerian path is an Eulerian circuit (you return to the starting point). If there are 2 odd-degree vertices, the Eulerian path will start and end at different vertices. If there are more than 2 odd-degree vertices, no Eulerian path exists.

Steps to solve:

1. Determine the degree (number of edges) for each vertex:

   • Vertex A: Degree 1 (one bridge).
   • Vertex B: Degree 3 (three bridges).
   • Vertex C: Degree 5 (five bridges).
   • Vertex D: Degree 4 (four bridges).
   • Vertex E: Degree 3 (three bridges).

2. Check for odd-degree vertices:

   • A has degree 1 (odd).
   • B has degree 3 (odd).
   • C has degree 5 (odd).
   • D has degree 4 (even).
   • E has degree 3 (odd).

   There are 4 odd-degree vertices (A, B, C, E).

Conclusion:

Since there are more than 2 vertices with odd degrees, it is impossible to find an Eulerian path that crosses each bridge exactly once. Therefore, the answer is DNE (Does Not Exist).

Would you like further details or explanations?

Here are 5 related questions:

1. What is an Eulerian circuit, and how does it differ from an Eulerian path?
2. Can the number of bridges affect whether an Eulerian path exists?
3. How do we apply graph theory to real-world problems like this one?
4. Is it possible to modify the city layout to make an Eulerian path possible?
5. How can we identify Hamiltonian paths in similar graphs?

Tip: Always count the degree of each vertex when dealing with Eulerian path problems; it's the key to determining whether a solution exists!