Dominating Set By Eric Wengert
Dominating Set Set of vertices such that every vertex is either in set S or adjacent to a vertex in S
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
Minimal Dominating Set There exists no vertex w such that S−{w} is also a dominating set in G
Dominating Set Decision Problem Given a graph G and an integer k Does G have a dominating set of size at most k?
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
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
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
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
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
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
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
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
References Proof https://www.youtube.com/watch?v=ba6HGbxSg1g https://math.stackexchange.com/questions/80721/vertex-cover-reduction https://stackoverflow.com/questions/5313919/proof-that-dominating-set-is-np-complete Algorithm https://disco.ethz.ch/courses/podc_allstars/lecture/chapter26.pdf