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Give step-by-step solution with explanation and final answer: Here are the minimum-spanning trees (MSTs) for Problems 1–6 using **Kruskal’s algorithm**. For each graph I sort edges by weight, then add the lightest edge that doesn’t make a cycle until the tree has $n-1$ edges. --- ### 1. **Edges (↑):** 1.2 DE, 1.8 BC, 2.2 BF, 2.3 CF, 2.4 BD, 2.9 AD, 3.0 AB, 3.6 BE, 4.1 EF, 4.7 DF **Pick:** DE, BC, BF, (skip CF—cycle), BD, AD → **MST = {DE, BC, BF, BD, AD}** **Total weight:** 1.2+1.8+2.2+2.4+2.9 = **10.5** --- ### 2. **Edges (↑):** 8 AD, 12 AC, 14 DF, 15 BD, 19 CD, 21 AB, 22 CE, 27 EF, 31 ED **Pick:** AD, AC, DF, BD, (skip CD—cycle), (skip AB—cycle), CE → **MST = {AD, AC, DF, BD, CE}** **Total weight:** 8+12+14+15+22 = **71** --- ### 3. **Edges (↑):** 1.8 BC, 2.1 AH, 2.3 BH, 2.7 EF, 2.8 AE, 2.9 CH, 3.0 AB, 3.5 EH, 3.9 CG, 4.1 GH, 4.2 HF, 4.8 FG **Pick:** BC, AH, BH, EF, AE, CG → **MST = {BC, AH, BH, EF, AE, CG}** **Total weight:** 1.8+2.1+2.3+2.7+2.8+3.9 = **15.6** --- ### 4. **Edges (↑):** 12 AD, 13 EF, 14 BC, 18 CF, 19 AB, 20 GH, 22 DG, 27 DE, 29 EH, 31 BE, 33 HI, 49 FI **Pick:** AD, EF, BC, CF, AB, GH, DG, HI → **MST = {AD, EF, BC, CF, AB, GH, DG, HI}** (**Skipped** DE/EH/BE as cycles once connected.) **Total weight:** 12+13+14+18+19+20+22+33 = **151** --- ### 5. **Edges (↑):** 21 BD, 23 AB, 28 BE, 29 AE, 30 AD, 34 AC, 36 BC, 43 CE, 44 DE, 50 CD **Pick:** BD, AB, BE, AC (skip AE/AD—cycles) → **MST = {BD, AB, BE, AC}** **Total weight:** 21+23+28+34 = **106** --- ### 6. **Edges (↑):** 0.9 CA, 1.9 GE, 2.2 CG, 2.7 AD, 3.2 BC, 3.4 BA, 3.7 BG, 4.6 CE, 5.4 ED, 6.0 EF, 7.4 DF **Pick:** CA, GE, CG, AD, BC, EF → **MST = {CA, GE, CG, AD, BC, EF}** **Total weight:** 0.9+1.9+2.2+2.7+3.2+6.0 = **16.9** --- If you’d like me to also do **Problem 13** (bridges), tell me how the four areas are labeled (or share a clearer map), and I’ll run Kruskal on that network too.

Question:

Give step-by-step solution with explanation and final answer: Here are the minimum-spanning trees (MSTs) for Problems 1–6 using **Kruskal’s algorithm**. For each graph I sort edges by weight, then add the lightest edge that doesn’t make a cycle until the tree has $n-1$ edges. --- ### 1. **Edges (↑):** 1.2 DE, 1.8 BC, 2.2 BF, 2.3 CF, 2.4 BD, 2.9 AD, 3.0 AB, 3.6 BE, 4.1 EF, 4.7 DF **Pick:** DE, BC, BF, (skip CF—cycle), BD, AD → **MST = {DE, BC, BF, BD, AD}** **Total weight:** 1.2+1.8+2.2+2.4+2.9 = **10.5** --- ### 2. **Edges (↑):** 8 AD, 12 AC, 14 DF, 15 BD, 19 CD, 21 AB, 22 CE, 27 EF, 31 ED **Pick:** AD, AC, DF, BD, (skip CD—cycle), (skip AB—cycle), CE → **MST = {AD, AC, DF, BD, CE}** **Total weight:** 8+12+14+15+22 = **71** --- ### 3. **Edges (↑):** 1.8 BC, 2.1 AH, 2.3 BH, 2.7 EF, 2.8 AE, 2.9 CH, 3.0 AB, 3.5 EH, 3.9 CG, 4.1 GH, 4.2 HF, 4.8 FG **Pick:** BC, AH, BH, EF, AE, CG → **MST = {BC, AH, BH, EF, AE, CG}** **Total weight:** 1.8+2.1+2.3+2.7+2.8+3.9 = **15.6** --- ### 4. **Edges (↑):** 12 AD, 13 EF, 14 BC, 18 CF, 19 AB, 20 GH, 22 DG, 27 DE, 29 EH, 31 BE, 33 HI, 49 FI **Pick:** AD, EF, BC, CF, AB, GH, DG, HI → **MST = {AD, EF, BC, CF, AB, GH, DG, HI}** (**Skipped** DE/EH/BE as cycles once connected.) **Total weight:** 12+13+14+18+19+20+22+33 = **151** --- ### 5. **Edges (↑):** 21 BD, 23 AB, 28 BE, 29 AE, 30 AD, 34 AC, 36 BC, 43 CE, 44 DE, 50 CD **Pick:** BD, AB, BE, AC (skip AE/AD—cycles) → **MST = {BD, AB, BE, AC}** **Total weight:** 21+23+28+34 = **106** --- ### 6. **Edges (↑):** 0.9 CA, 1.9 GE, 2.2 CG, 2.7 AD, 3.2 BC, 3.4 BA, 3.7 BG, 4.6 CE, 5.4 ED, 6.0 EF, 7.4 DF **Pick:** CA, GE, CG, AD, BC, EF → **MST = {CA, GE, CG, AD, BC, EF}** **Total weight:** 0.9+1.9+2.2+2.7+3.2+6.0 = **16.9** --- If you’d like me to also do **Problem 13** (bridges), tell me how the four areas are labeled (or share a clearer map), and I’ll run Kruskal on that network too.

This is AI generated Response.
Asked by: Nehal Anurag
Created at: 2025-09-04 10:38:29

Final Answer:

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Let's break down how to find the minimum spanning tree (MST) using **Kruskal's algorithm** with detailed steps and explanations, using one of your problems as a model. I'll do **Problem 1** step-by-step. If you want another, let me know! --- ## **Problem 1: Step-by-step Solution with Explanations** ### **Given Edges (sorted by weight):** | Edge | Weight | |------|--------| | DE | 1.2 | | BC | 1.8 | | BF | 2.2 | | CF | 2.3 | | BD | 2.4 | | AD | 2.9 | | AB | 3.0 | | BE | 3.6 | | EF | 4.1 | | DF | 4.7 | Assume the vertices are **A, B, C, D, E, F**. --- ### **Step 1: Sort all edges by weight (already sorted above).** ### **Step 2: Initialize MST as empty.** We want to select edges one by one, lightest first, **as long as they don't form a cycle**. We need **n-1 = 6-1 = 5** edges for a tree of 6 vertices. --- ### **Step 3: Add edges one at a time, checking for cycles.** #### **Current MST = ∅** #### **Edge 1: DE (1.2)** - **Vertices D and E are not connected yet.** - Add DE to MST. > MST = {DE} #### **Edge 2: BC (1.8)** - **Vertices B and C are not connected yet.** - Add BC to MST. > MST = {DE, BC} #### **Edge 3: BF (2.2)** - **Vertices B and F are not connected yet.** - Add BF to MST. > MST = {DE, BC, BF} #### **Edge 4: CF (2.3)** - **Check for cycle:** - B is connected to C via BC, and B is connected to F via BF. - So C-B-F are all connected. - CF would connect C and F, which are already connected via B. - **Adding CF would form a cycle. Skip.** #### **Edge 5: BD (2.4)** - **B is in group {B, C, F}** - **D is in group {D, E}** - **No cycle.** Add BD. > MST = {DE, BC, BF, BD} #### **Edge 6: AD (2.9)** - **D is now in group {B, C, F, D, E}** - **A is not yet connected to anything.** - **No cycle.** Add AD. > MST = {DE, BC, BF, BD, AD} Now the MST has 5 edges (for 6 vertices). **We are done!** --- ### **Step 4: List MST edges and calculate total weight** - MST = {DE, BC, BF, BD, AD} - Total weight = 1.2 + 1.8 + 2.2 + 2.4 + 2.9 = **10.5** --- ### **Step 5: Write Final Answer** **Minimum Spanning Tree (MST) using Kruskal's algorithm:** \[ \text{MST} = \{DE, BC, BF, BD, AD\} \] \[ \text{Total weight} = 1.2 + 1.8 + 2.2 + 2.4 + 2.9 = \boxed{10.5} \] --- ### **Summary Table of Steps** | Step | Picked Edge | Cycle? | MST Edges So Far | Total Weight | |------|-------------|--------|-------------------------|--------------| | 1 | DE (1.2) | No | DE | 1.2 | | 2 | BC (1.8) | No | DE, BC | 3.0 | | 3 | BF (2.2) | No | DE, BC, BF | 5.2 | | 4 | CF (2.3) | Yes | DE, BC, BF | 5.2 | | 5 | BD (2.4) | No | DE, BC, BF, BD | 7.6 | | 6 | AD (2.9) | No | DE, BC, BF, BD, AD | 10.5 | --- If you'd like the **step-by-step process for any other problem**, just specify the number!

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