To solve this problem, we need to:

1. Eulerize the graph: This involves duplicating the minimum number of edges to make all vertices have an even degree (number of edges connected to a vertex). A graph has an Eulerian circuit if all vertices have even degrees, and it is connected.

2. Find an Eulerian circuit: Once the graph is Eulerized, we will traverse all edges exactly once, starting and ending at the same vertex.

Step 1: Analyzing the degrees of vertices

From the graph, let’s determine the degree of each vertex:

• $A$: degree 3 (connected to $B$, $E$, $D$)
• $B$: degree 4 (connected to $A$, $C$, $D$, $E$)
• $C$: degree 3 (connected to $B$, $D$, $E$)
• $D$: degree 3 (connected to $A$, $B$, $C$)
• $E$: degree 3 (connected to $A$, $B$, $C$)

Since only vertex $B$ has an even degree, we need to add edges to the other vertices to make their degrees even. The best approach is to add the minimum number of edges to achieve this.

Step 2: Eulerizing the graph

We need to duplicate edges to make the degrees of $A$, $C$, $D$, and $E$ even. Here's the minimal solution:

• Duplicate edge $A-D$, making $A$ and $D$ have degree 4.
• Duplicate edge $C-E$, making $C$ and $E$ have degree 4.

Now, all vertices have even degrees:

• $A$: degree 4
• $B$: degree 4
• $C$: degree 4
• $D$: degree 4
• $E$: degree 4

Step 3: Finding the Eulerian circuit

Now that all vertices have even degrees, the graph is Eulerian. Starting at vertex $A$, one possible Eulerian circuit is: $A - D - C - E - B - A - E - C - D - B$

So, the Euler circuit is: $ADCBEACEBD$

Would you like more details on the Eulerization process or on how Euler circuits are constructed? Here are 5 related questions you might be interested in:

1. What conditions must a graph meet to have an Eulerian circuit?
2. How is an Euler path different from an Euler circuit?
3. Can all graphs be Eulerized by duplicating edges, or are there exceptions?
4. What is the difference between Eulerian and Hamiltonian circuits?
5. How does edge duplication in Eulerization affect the structure of the graph?

Tip: When Eulerizing a graph, always focus on the vertices with odd degrees—adding edges between them is the key to making the graph Eulerian.