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I am writing an algorithm to find the dominating set of a tournament graph. Is the minimum spanning tree of a directed graph equivalent to the dominating set of the graph? In other words, if I find the smallest MST for the tournament graph (by iterating through all of the vertices), can I then say this is equivalent to the dominating set of the graph?

Mat
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el diablo
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  • 'smallest' is redundant in 'smallest MST'. MST is smallest by definition. – jfs Nov 17 '08 at 18:40
  • Tournament Dominating Set is [LogSNP-complete](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:L#logsnp) and furthermore [W\[2\]-complete](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:W#w1). It is unlikely that there is a polynomial time algorithm for Tournament Dominating Set. However, MST is solvable in polynomial time. Furthermore, a universal vertex is a dominating set, whereas an MST always has all vertices. – Pål GD Apr 29 '15 at 01:14

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This Wikipedia article states that the problems of finding a dominating set and finding a spanning tree are equivalent. Given a spanning tree, the non-leaf nodes form a dominating set, and given a connected dominating set, you can easily obtain of the original graph joining one spanning tree of it with the vertexes that do not belong to it. However, finding a minimum spanning tree and finding a minimal dominating set are different problems. I guess that they are equivalent again, but I'm not sure.

Federico A. Ramponi
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    Not equivalent - Minimum Dominating Set is NP complete, Minimum spanning tree is solvable by Prim's or Kruskal's algorithm, at a cost of O(V^2) – Thelema May 17 '10 at 03:42
  • @Thelema Tournament Dominating Set is not NP-complete, but LogSNP-complete. – Pål GD Apr 29 '15 at 01:18
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No, because the MST will include all vertices of the graph, and the dominating set might not.

See for example the graph here: http://en.wikipedia.org/wiki/Tournament_(graph_theory) Vertices 2 and 4 create a dominating set, and not a spanning tree.

jfs
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DanJ
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  • I have not yet figured out how to get Wikipedia URLs with parentheses to come out properly in SO. Any suggestions? – sep332 Nov 17 '08 at 18:44