1. Dominating Set
By Eric Wengert

2. Dominating Set
Set of vertices such that every vertex is either in set S or adjacent to a vertex in S

3. Dominating Set
Set of vertices such that every vertex is either in set S or adjacent to a vertex in S
Maximum Independent set is a dominating set

4. Minimal Dominating Set
There exists no vertex w such that S−{w} is also a dominating set in G

5. Dominating Set Decision Problem
Given a graph G and an integer k
Does G have a dominating set of size at most k?

6. DS in NP For each vertex in graph G Success
Check if vertex is in S or adjacent to a vertex in S
If vertex not in S and not adjacent to S, reject
Success
Can be done in polynomial time
Therefore dominating set is in NP

7. DS in NP-complete Vertex cover is NP-Complete
Vertex cover to dominating set
K for VC = 3
K for DS = 2
Change graph so k for DS = 3

8. DS in NP-complete Vertex cover to dominating set
Add vertices and edges to make triangles
Claim: 𝐺 ′ has a DS of size at most k iff G has a vertex cover of size at most k

9. DS in NP-complete Vertex cover to dominating set
Add vertices and edges to make triangles
Find degree of original nodes 6 2 4 4 4 4

10. DS in NP-complete Vertex cover to dominating set
Add vertices and edges to make triangles
Find degree of original nodes
Greedy: choose vertex with highest degree 6 2 4 4 4 4

11. DS in NP-complete Vertex cover to dominating set
Add vertices and edges to make triangles
Find degree of original nodes
Greedy: choose vertex with highest degree 6 2 4 4 4 4

12. DS in NP-complete Vertex cover to dominating set
Add vertices and edges to make triangles
Find degree of original nodes
Greedy: choose vertex with highest degree
One vertex left over so DS k = 3 6 2 4 4 4 4

13. Greedy Algorithm
S = Ø
While there are vertices not in or adjacent to S
v = {v | w(v) = maxu {w(u)}} // w(v) is degree
S = S U v
ln(Δ) approximation // Δ is the maximal degree of G

14. References Proof https://www.youtube.com/watch?v=ba6HGbxSg1g
Algorithm