1. Kruskal’s Algorithm
AQR

2. Kruskal’s Algorithm
Assume that you start with a table of the weights associated with each edge (just like the Railroad problem in Student Activity Sheet 7).
Step 1: Put all of the weights in a list from smallest to largest.
Step 2: Find the smallest weight in the list and include the associated edge and two vertices, as long as that does not create a cycle.
Step 3: Remove this weight from the list.
Step 4: Repeat Steps 2 and 3 until all vertices are connected.

3. Kruskal’s Algorithm
2, 4, 5, 7, 9, 10, 11, 12 AE AD DC BC (BD would make a circuit so take it of but do not add) And so on

4. Kruskal’s Algorithm Work with edges, rather than nodes Two steps:
Sort edges by increasing edge weight
Select the first |V| – 1 edges that do not generate a cycle

5. Walk-Through F C A B D H G E Consider an undirected, weight graph 3 104 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

6. F C A B D H G E Sort the edges by increasing edge weight edge dv (D,E)3 F C
edge dv
(D,E) 1
(D,G) 2
(E,G) 3
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

7. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

8. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

9. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3
Accepting edge (E,G) would create a cycle

10. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

11. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

12. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

13. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

14. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

15. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

16. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

17. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5 
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

18. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5 
(D,F) 6
(A,B) 8
(A,F) 10 A 3 4 5 B D H 1 2 } 3 G E
not
considered
Done
Total Cost =  dv = 21