1. **Problem Statement:** Find the shortest eligible walk in the given graph with vertices V1, V2, V3, V4 and edges e1 (V1-V2), e2 (V1-V3), e3 (V1-V4), e4 (V3-V4). The shortest walk is the one with the fewest vertices counted with repetition. 2. **Understanding Walks:** A walk is a sequence of vertices and edges where each edge connects the vertices before and after it. Vertices can be repeated. 3. **Analyze Each Option:** - Option 1: V1 e1 V2 e1 V1 has vertices V1, V2, V1 (3 vertices). - Option 2: V1 V3 V4 V1 has vertices V1, V3, V4, V1 (4 vertices). - Option 3: V4 e4 V3 e2 V1 e1 V2 has vertices V4, V3, V1, V2 (4 vertices). - Option 4: V2 e1 V3 is invalid because e1 connects V1 and V2, not V2 and V3. 4. **Check Validity:** Option 4 is invalid. 5. **Compare Lengths:** Option 1 has 3 vertices, options 2 and 3 have 4 vertices. 6. **Conclusion:** The shortest eligible walk is Option 1: V1 e1 V2 e1 V1. **Final answer:** V1 e1 V2 e1 V1