Let's solve your problem in **2 steps** as requested: --- ### **Step 1: Interpret the Problem and Find Adjacencies** The goal is to place security cameras in rooms so that every room is monitored, and each camera can monitor the room it is installed in **plus all rooms directly connected by a door** (adjacent rooms). #### **Room Adjacency List from the Diagram:** - **Room 1:** Room 2, Room 4 - **Room 2:** Room 1, Room 3, Room 5 - **Room 3:** Room 2, Room 6 - **Room 4:** Room 1, Room 5, Room 7 - **Room 5:** Room 2, Room 4, Room 6, Room 8 - **Room 6:** Room 3, Room 5, Room 9 - **Room 7:** Room 4, Room 8 - **Room 8:** Room 5, Room 7, Room 9 - **Room 9:** Room 6, Room 8 **Explanation:** Each list shows which rooms are directly adjacent (connected by a door), which is important because a camera in one room can monitor that room and all its adjacent rooms. --- ### **Step 2: Find the Minimum Number of Cameras and Their Locations** This is a version of the "minimum dominating set" problem for the given graph. #### **Let's try to cover all rooms with the minimum number of cameras:** - **Try putting cameras in Room 5 (center), Room 2, and Room 8:** - **Camera in Room 5:** Covers Rooms 2, 4, 5, 6, 8 - **Camera in Room 2:** Covers Rooms 1, 2, 3, 5 - **Camera in Room 8:** Covers Rooms 5, 7, 8, 9 Now, let's check coverage: - Room 1: covered by Room 2's camera - Room 2: covered by Room 2's camera and Room 5's camera - Room 3: covered by Room 2's camera - Room 4: covered by Room 5's camera - Room 5: covered by all three cameras - Room 6: covered by Room 5's camera - Room 7: covered by Room 8's camera - Room 8: covered by Room 8's camera and Room 5's camera - Room 9: covered by Room 8's camera **All rooms are covered with cameras in Room 2, Room 5, and Room 8 (3 cameras total).** **Explanation:** By carefully selecting rooms that are highly connected (especially the center, Room 5), we can minimize the number of cameras. Checking each room's coverage ensures we don't miss any. --- ### **Final Answer** - **The minimum number of cameras needed is:** **3** - **They should be placed in:** **Room 2, Room 5, and Room 8**

Let's proceed step-by-step to determine the minimum number of security cameras needed and their optimal placement. --- ### **Step 1: Understand the graph and the problem** **Explanation:** The diagram shows 9 rooms connected by doors. Placing a camera in a room covers that room plus all rooms directly connected to it. Our goal is to cover all rooms with the minimum number of cameras. **Adjacency list based on the diagram:** - Room 1: 2, 4 - Room 2: 1, 3, 5 - Room 3: 2, 6 - Room 4: 1, 5, 7 - Room 5: 2, 4, 6, 8 - Room 6: 3, 5, 9 - Room 7: 4, 8 - Room 8: 5, 7, 9 - Room 9: 6, 8 --- ### **Step 2: Find the minimum set of rooms for cameras** **Explanation:** We look for a set of rooms such that every room is either in this set or adjacent to a room in this set. This is a typical "dominating set" problem. **Candidate placement:** - Place cameras in **Room 5** (center), which covers Rooms 2, 4, 5, 6, 8. - To cover Rooms 1 and 3 (adjacent to Room 2), place a camera in **Room 2**. - To cover Room 7 and 9 (adjacent to Rooms 4 and 8), place a camera in **Room 8**. **Coverage check:** - **Room 1:** covered by Room 2 - **Room 2:** camera itself - **Room 3:** covered by Room 2 - **Room 4:** covered by Room 5 - **Room 5:** camera itself - **Room 6:** covered by Room 5 - **Room 7:** covered by Room 8 - **Room 8:** camera itself - **Room 9:** covered by Room 8 **Total cameras:** 3 (Rooms 2, 5, and 8) --- ### **Final answer:** **The minimum number of cameras needed is 3, placed in Rooms 2, 5, and 8.**