📍 Scene Setup
The school's network hub has crashed overnight, displaying NODE ZERO ERROR across every screen. Josh has been handed the original network design specs and must verify each parameter before the system can reboot.
Every server is locked behind a topology check. Six verifications stand between Josh and a working network.
📋 Key Formulas, Graph Theory
Handshaking Lemma
sum of degrees = 2 × edges
Tree formula
edges = n − 1 vertices
Planar graphs
v − e + f = 2
Degree = number of edges connected to a vertex · Tree = connected, no cycles, n−1 edges
1
Server 1, Handshaking Verification
The main switch logs show 7 cable connections (edges) running through it. What is the sum of all port-connection counts (sum of all vertex degrees) across the entire network segment?
Use the Handshaking Lemma, every cable connects exactly two ports.
💡 Sum of degrees = 2 × number of edges. Here that's 2 × 7 = ?
✅ Sum = 14. Every cable contributes 2 to the total count, one for each end.
2
Server 2, Backup Tree Specs
The backup network is a tree (no redundant paths) with exactly 13 nodes. How many cable connections does it need?
Trees always use the minimum possible edges to keep everything connected.
💡 Tree formula: edges = n − 1. Here n = 13, so edges = 13 − 1 = ?
✅ 12 cables. A tree always uses n − 1 edges for n nodes, no more, no less.
3
Server 3, Hub Degree Check
The central hub HQ-1 has direct cable links to exactly five devices: Alpha, Beta, Gamma, Delta, and Epsilon. What is the degree of HQ-1?
Degree = number of edges directly connected to a vertex.
💡 Count the devices listed: Alpha, Beta, Gamma, Delta, Epsilon. How many is that?
✅ Degree 5. HQ-1 is connected to 5 other devices, so it has 5 edges.
4
Server 4, Sector Edge Count
A network sector has 6 devices. The number of connections at each device (degrees) are: 4, 2, 3, 1, 3, 3. How many total cable connections (edges) are in this sector?
✅ 8 cables. Sum of degrees = 16, so edges = 16 ÷ 2 = 8.
5
Server 5, Minimal Tree Design
Josh needs to design a new section of the network as a tree: fully connected, no cycles, minimum cables possible. It has 7 nodes. How many edges does it need?
Trees are the most efficient connected structure, no redundancy at all.
💡 Tree formula: edges = n − 1 = 7 − 1 = ?
✅ 6 edges. Seven nodes, six cables, enough to connect everything with zero redundancy.
6
Server 6, Planar Map Regions
The final network segment is drawn as a planar diagram (no crossing lines). It has 7 nodes and 9 connections. Using the formula v − e + f = 2, how many regions does the diagram divide the page into?
Regions (faces) include the unbounded outer region. Remember to include it in your count.
💡 v − e + f = 2 → 7 − 9 + f = 2 → f = 2 − 7 + 9 = ?
✅ 4 faces. Using v − e + f = 2: 7 − 9 + f = 2 → f = 4. Network fully verified!
🌐
Network Restored!
All six servers are back online. Josh's graph theory diagnostics were perfect, every topology parameter checked out and the network rebooted without a hitch.