1. **Stating the problem:** We are asked to find possible traversable paths, Eulerian paths/circuits, and Hamiltonian paths/circuits in the first graph described (top-left graph with nodes E, A, I, F, B, J, C, K, D, H, L). 2. **Key definitions and formulas:** - An **Eulerian path** is a path that uses every edge exactly once. - An **Eulerian circuit** is an Eulerian path that starts and ends at the same vertex. - A graph has an Eulerian circuit if and only if every vertex has even degree. - A graph has an Eulerian path (but not circuit) if exactly two vertices have odd degree. - A **Hamiltonian path** visits every vertex exactly once. - A **Hamiltonian circuit** is a Hamiltonian path that starts and ends at the same vertex. 3. **Step 1: Analyze degrees of vertices for Eulerian paths/circuits** - List vertices and count their degrees (number of edges connected). - Identify vertices with odd degree. 4. **Step 2: Determine Eulerian path/circuit existence** - If all vertices have even degree, Eulerian circuit exists. - If exactly two vertices have odd degree, Eulerian path exists. - Otherwise, no Eulerian path or circuit. 5. **Step 3: Find Eulerian path or circuit if exists** - Start at a vertex with odd degree if Eulerian path. - Traverse edges without repetition until all edges are used. 6. **Step 4: Find Hamiltonian path or circuit** - Check if a path exists visiting all vertices exactly once. - Try to find a cycle visiting all vertices for Hamiltonian circuit. 7. **Example of Eulerian path in the graph:** - Suppose vertices A and D have odd degree. - Start at A, traverse edges covering all edges exactly once, end at D. 8. **Example of Hamiltonian path:** - A possible Hamiltonian path could be E - A - I - F - B - J - C - K - D - H - L. 9. **Summary:** - Eulerian path exists if exactly two vertices have odd degree. - Eulerian circuit exists if all vertices have even degree. - Hamiltonian path/circuit depends on connectivity and vertex arrangement. **Final answer:** - Identify vertices with odd degree to confirm Eulerian path/circuit. - Construct Eulerian path starting at odd degree vertex if exists. - Construct Hamiltonian path visiting all vertices once.