To determine whether it is possible to traverse all the bridges in the city exactly once (an Eulerian path), we need to analyze the graph structure based on the given diagram.

Steps:

1. Create a Graph Representation:

   • Vertices: $A, B, C, D, E$ (the landmasses)
   • Edges: Bridges connecting these vertices.

   From the image:

   • $A$ connects to $C$.
   • $B$ connects to $C$.
   • $C$ connects to $D$.
   • $D$ connects to $E$.

2. Check the Degree of Each Vertex: For an Eulerian path to exist, the graph must have either:

   • Exactly 0 vertices with an odd degree (Eulerian circuit), or
   • Exactly 2 vertices with an odd degree (Eulerian path).

   Calculate degrees:

   • Vertex $A$: 1 bridge → Degree = 1 (odd)
   • Vertex $B$: 1 bridge → Degree = 1 (odd)
   • Vertex $C$: 3 bridges → Degree = 3 (odd)
   • Vertex $D$: 3 bridges → Degree = 3 (odd)
   • Vertex $E$: 1 bridge → Degree = 1 (odd)

3. Conclusion: There are more than two vertices ($A, B, C, D, E$) with an odd degree. Therefore, it is not possible to find an Eulerian path through the graph.

Final Answer:

DNE (Does Not Exist)

Would you like further details on Eulerian paths, graph theory concepts, or a breakdown of the logic used here?

Related Questions:

1. What are Eulerian paths, and how do they differ from Eulerian circuits?
2. Can a graph with all vertices having an even degree always have an Eulerian circuit?
3. How do you determine if a graph is connected before checking for Eulerian properties?
4. What modifications to the current graph would make an Eulerian path possible?
5. How do bridge problems like this apply to real-life city planning?

Tip: Eulerian paths are a foundational concept in graph theory and are frequently used in logistics and route planning.