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Dominating Set By Eric Wengert
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Dominating Set Set of vertices such that every vertex is either in set S or adjacent to a vertex in S
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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
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Minimal Dominating Set
There exists no vertex w such that S−{w} is also a dominating set in G
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Dominating Set Decision Problem
Given a graph G and an integer k Does G have a dominating set of size at most k?
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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
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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
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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
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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
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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
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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
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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
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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
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References Proof https://www.youtube.com/watch?v=ba6HGbxSg1g
Algorithm
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