Dominating Set By Eric Wengert Dominating Set Set

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 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 §

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

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

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

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

References Proof § https: //www. youtube. com/watch? v=ba 6 HGbx. Sg 1 g § https: //math. stackexchange. com/questions/80721/vertexcover-reduction § https: //stackoverflow. com/questions/5313919/proof-thatdominating-set-is-np-complete Algorithm § https: //disco. ethz. ch/courses/podc_allstars/lecture/chapter 26. pdf