2. [6 marks] Consider the following floor-plan of a house: 

Lounge Hall Kitchen Dining room Conservatory Garden The owner, Ingvar, has forgotten to build any interior doors. 
He would like to spend as little as possible, and so wants to install the minimum number of doors that will make his house completely accessible. 
He would also like visitors to his house to be able to visit each room in turn, starting from the hall, with a minimum amount of backtracking. 
Use the graph theory you have learnt in this module to solve Ingvar’s problem. In your answer, you should: 

(a) [1 mark] Convert the floor plan into a graph, with one node 𝐷, 𝐻, 𝐾, 𝐿, 𝐶, 𝐺 for each of the six rooms, and an edge between two rooms if they share a wall. 

(b) [1 mark] State which graph-theoretic concept will provide a solution to the problem of installing a minimum number of doors that will make the house completely accessible. 

(c) [1 mark] State an algorithm studied in the unit which will provide a solution which satisfies Ingvar’s constraints. 

(d) [3 marks] Carry out this algorithm on the graph, and illustrate your answer on a copy of the original floor-plan.

Let's approach the problem step-by-step:

### (a) Converting the floor plan into a graph:
The floor plan can be represented as a graph where:
- **Nodes** represent the rooms: Hall (H), Lounge (L), Kitchen (K), Dining room (D), Conservatory (C), and Garden (G).
- **Edges** represent walls between rooms where doors can be placed.

Assuming the rooms share walls as follows:
- Hall (H) is adjacent to the Lounge (L), Kitchen (K), and Dining room (D).
- The Dining room (D) is adjacent to the Conservatory (C).
- The Conservatory (C) is adjacent to the Garden (G).
- The Kitchen (K) and Lounge (L) are not directly connected to each other.
  
Thus, the graph would look like this:
- Nodes: {H, L, K, D, C, G}
- Edges: 
  - (H, L)
  - (H, K)
  - (H, D)
  - (D, C)
  - (C, G)

### (b) Graph-theoretic concept:
The graph-theoretic concept needed to solve this problem is the **Minimum Spanning Tree (MST)**. The MST ensures that all rooms (nodes) are connected with the minimum number of edges (doors) while keeping the house accessible.

### (c) Algorithm:
The algorithm to solve this problem is **Kruskal’s Algorithm** or **Prim’s Algorithm**, both of which are used to find a Minimum Spanning Tree (MST). These algorithms ensure that the graph is connected with the least number of edges.

### (d) Applying Kruskal’s Algorithm:

**Step 1: Sort edges by weight (assuming equal weight for each wall).**  
Since all walls are equal in this problem, we can choose any valid edge set.

**Step 2: Add edges while avoiding cycles.**
We start with no edges and progressively add edges from the sorted list, making sure not to create any cycles, as that would indicate backtracking.

- Add (H, L), since there are no edges yet.
- Add (H, K), no cycle formed.
- Add (H, D), still no cycle.
- Add (D, C), no cycle formed.
- Add (C, G), the graph remains acyclic and connected.

Thus, the minimum spanning tree includes edges (H, L), (H, K), (H, D), (D, C), (C, G).

**Illustrating the answer:**
In the final floor plan:
- A door should be placed between Hall and Lounge.
- A door should be placed between Hall and Kitchen.
- A door should be placed between Hall and Dining room.
- A door should be placed between Dining room and Conservatory.
- A door should be placed between Conservatory and Garden.

This results in a fully connected house with the minimum number of doors and no backtracking, allowing visitors to traverse the house efficiently.

Would you like a detailed explanation of any of these steps?  

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### 5 Related Questions:
1. How would Prim’s Algorithm work differently compared to Kruskal’s Algorithm in this scenario?
2. Can you describe what a cycle in a graph represents in terms of the house's floor plan?
3. How would the problem change if some rooms shared multiple walls?
4. What is the significance of minimizing edges in a Minimum Spanning Tree?
5. How would you adapt the solution if there were extra costs associated with different doors?

### Tip:
In graph theory, always remember that an MST connects all nodes with the least possible edges, which is crucial in minimizing costs or traversal efforts.

2. [6 marks] Consider the following floor-plan of a house: Lounge Hall Kitchen Dining room Conservatory Garden. The owner, Ingvar, has forgotten to build any interior doors. He would like to spend as little as possible, and so wants to install the minimum number of doors that will make his house completely accessible. He would also like visitors to his house to be able to visit each room in turn, starting from the hall, with a minimum amount of backtracking. Use the graph theory you have learnt in this module to solve Ingvar’s problem.

Discover how to use graph theory to solve a real-world problem involving a house floor plan. Learn how to convert a floor plan into a graph and apply algorithms like Kruskal's and Prim's to find the Minimum Spanning Tree (MST) that minimizes door installation while ensuring accessibility. Suitable for undergraduate students.

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