Let's **explain** and **solve** this problem using the **Minimal Spanning Tree (MST)** concept. --- ### 1. **Explanation** A **Minimal Spanning Tree (MST)** of a graph is a subset of the edges that connects all the vertices together, without any cycles, and with the minimum possible total edge weight. Here, each "activity" is an edge between two nodes (vertices), and "duration" is the weight (cost) of that edge. Common algorithms to find MST: - **Kruskal's Algorithm** - **Prim's Algorithm** We'll use **Kruskal's Algorithm** for its simplicity with edge lists. --- ### 2. **Steps (Kruskal's Algorithm)** #### **Step 1: List all edges in increasing order of duration** | Edge | Duration | |------|----------| | 6-7 | 2 | | 8-9 | 2 | | 1-3 | 3 | | 9-10 | 3 | | 1-2 | 4 | | 2-4 | 4 | | 1-5 | 5 | | 3-6 | 6 | | 4-7 | 6 | | 6-8 | 9 | | 7-9 | 8 | | 5-6 | 10 | #### **Step 2: Add edges one by one (smallest first), avoid cycles, until all nodes are connected** Let's track which nodes are connected as we go: 1. **6-7 (2)** → add 2. **8-9 (2)** → add 3. **1-3 (3)** → add 4. **9-10 (3)** → add 5. **1-2 (4)** → add 6. **2-4 (4)** → add 7. **1-5 (5)** → add 8. **3-6 (6)** → add - At this point, let's check the number of edges. - For 10 nodes, MST has 9 edges. So, let's list the edges added so far: | Edge | Duration | |------|----------| | 6-7 | 2 | | 8-9 | 2 | | 1-3 | 3 | | 9-10 | 3 | | 1-2 | 4 | | 2-4 | 4 | | 1-5 | 5 | | 3-6 | 6 | | 4-7 | 6 | - Now, let's check for cycles and connections: - All nodes: 1,2,3,4,5,6,7,8,9,10 are connected. - No cycles formed (no node is revisited in the same path). We have **9 edges** (for 10 nodes), so we stop here. --- ### 3. **Final Summary** **The minimal spanning tree connects all nodes with the minimum total duration:** #### **Edges in the MST:** - 6-7 (2) - 8-9 (2) - 1-3 (3) - 9-10 (3) - 1-2 (4) - 2-4 (4) - 1-5 (5) - 3-6 (6) - 4-7 (6) #### **Total Minimum Duration:** 2 + 2 + 3 + 3 + 4 + 4 + 5 + 6 + 6 = **35** --- ### **Final Answer** > The minimal spanning tree for the given network consists of the edges listed above, with a total minimal duration of **35**. This connects all activities using the least total duration without forming cycles.