Let's solve the problem step by step as requested: --- ### **Step 1: Introduction to the Concept** An **Euler circuit** in a connected graph is a circuit that visits every edge exactly once and returns to the starting vertex. For a connected graph to have an Euler circuit, **every vertex must have an even degree** (i.e., the number of edges at each vertex is even). --- ### **Step 2: Analyzing Degree Sequence** Given a connected graph \( G \) with degree sequence \( d_1, d_2, \ldots, d_n \), we identify the vertices with **odd degrees**. Let's say there are \( k \) vertices with odd degree in the graph. --- ### **Step 3: Edges Needed for Euler Circuit** To transform the graph so that **all vertices have even degree** (a requirement for an Euler circuit), we must pair up all vertices with odd degree. Each added edge will connect two odd-degree vertices, making both their degrees even. Thus, the number of edges needed is \( \frac{k}{2} \), where \( k \) is the number of odd-degree vertices. --- ### **Step 4: Final Answer (Concise)** **Final Answer:** To make the connected graph \( G \) have an Euler circuit, you must add \( \frac{k}{2} \) edges, where \( k \) is the number of vertices with odd degree in the degree sequence \( d_1, d_2, ..., d_n \). --- ### **Step 5: Summary of Steps** 1. Introduced Euler circuit and its condition. 2. Identified vertices with odd degrees. 3. Determined the number of edges needed to make all degrees even. 4. Gave a concise final answer. **In summary:** We explained the Euler circuit condition, counted odd-degree vertices, calculated required edges, and gave the final answer.