The problem you provided asks for an optimal Eulerization of the graph shown in Figure 5-57. Eulerization refers to making a graph Eulerian, which means ensuring that all vertices have an even degree by adding the fewest possible edges.

To solve this problem, let's follow these steps:

1. Check the degree of each vertex: First, calculate the degree (number of edges incident to a vertex) for each vertex in the graph. If any vertex has an odd degree, you will need to add edges to it.

2. Find the odd-degree vertices: In Euler's theorem, for a graph to have an Eulerian circuit (a path that uses every edge exactly once and returns to the starting point), all vertices must have an even degree. If the graph has any odd-degree vertices, they need to be paired, and additional edges will be added between them.

3. Pair the odd-degree vertices: After identifying the vertices with an odd degree, find the shortest path between them and add those edges to make all the degrees even.

4. Construct the Eulerian circuit: Once the graph has no more odd-degree vertices, an Eulerian circuit can be constructed, and the graph will be Eulerized.

Let me check the graph details and walk you through the degree of each vertex.It seems that I do not currently have the ability to directly interpret the graph you've uploaded. However, you can describe the structure or layout of the graph, and I can guide you step by step in determining the optimal Eulerization.

Here's how you can proceed:

1. Describe the graph: Provide a brief description of the vertices and edges (e.g., a grid with labeled vertices A, B, C, D, etc.).
2. Identify odd-degree vertices: Tell me which vertices have an odd degree, and I can help with pairing them and finding the optimal Eulerization.

Alternatively, you can take a screenshot of the degree of each vertex, and we can work from there! Let me know how you'd like to proceed.