I'd like to know the name and the algorithm for the following problem which I'm guessing is a classic, but is slightly different from graph connectivity.

Consider a undirected graph G=(V,E). What is the minimum number of vertices which removal (vertices and their adjacent edges) makes the resulting sub-graph empty of any edge?

For example: If A - B - C, then just need to remove B. If A - B - C and A - C, then need to remove at two vertices (any pair).

For an algo, intuitively I'd proceed by removing first the vertex of highest degree and proceed the same on the remaining graph until there are no edges. Not sure if it gives the min number. For sure, in the worst case I can always go through all possible |V|! removal possibilities and take the min.

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    $\begingroup$ Welcome to Computer Science! Note that you can use LaTeX here to typeset mathematics in a more readable way. See here for a short introduction. $\endgroup$ – FrankW Sep 16 '14 at 13:39
  • $\begingroup$ Can you clarify what you mean by "makes the resulting sub-graph empty of any edge?" Do you mean, the minimum number of vertices whose removal leaves a graph with no remaining edges? $\endgroup$ – D.W. Sep 16 '14 at 22:36

The problem you describe is called minimum vertex cover. It is a well-known NP-complete problem, so there is no known algorithm that will always be fast (as in polynomial runtime) and give the optimal solution. However, fast approximation solutions are known, as well as algorithms that, while slow in the worst case on general graphs, are fast on significant classes of graphs. See the linked Wikipedia article for details and links to even more details.

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  • $\begingroup$ Perfect, that article covers it. $\endgroup$ – jam123 Sep 16 '14 at 13:53
  • $\begingroup$ Are you aware of an (fast) approximation of the min vertex cover with some guarantees on the approximation cover being not too large? Ideally, a guarantee that the approximate cover is smaller than the min cover - which I'm guessing is not usually the aim of approximation, but in my case I prefer a smaller cover that might still leave a few edges. $\endgroup$ – jam123 Sep 16 '14 at 15:46
  • $\begingroup$ To clarify, I mean an approximation that returns either a minimum cover (ideal) or a smaller cover, not larger. $\endgroup$ – jam123 Sep 16 '14 at 15:54
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    $\begingroup$ I don't know one off the top of my head. But it would be perfectly appropriate (and sensible), if you asked this as a new question, if you can't find anything yourself. (Just make sure, the answer isn't easily found by using a search engine.) $\endgroup$ – FrankW Sep 16 '14 at 16:16

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