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You are given a Graph G and an integer k.

The question is: Does G contain a vertex set X of size at most k such that deleting X results in a graph of degree at most 1?

Design an FPT (or a kernelization) algorithm for this problem and argue why it is correct.

Hint: Observe that vertices of higher degree must be "dealt with" somehow.

I do not exactly know where to start. I suppose that some reduction rules can be applied first, but not so sure what those can be (probably something that "deals with" vertices of higher degree.

Any ideas?

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I'm not sure about the FPT algorithm, but I have an idea for the kernelization.

All vertices with degree > k + 1 must be in X, since otherwise you would have to pick all but one of its neighbours which would make |X| > k. So you can delete a vertex with degree > k + 1 and rerun the algorithm with the remaining graph and k = k - 1.

You can also delete any vertices with degree 0, as their deletion doesn't change the degree of any other vertices.

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  • $\begingroup$ Yes, you are right about that. You can delete vertices with zero degree and vertices with degree > k +1. What I cannot argue is "after the deletion of X, graph has a degree of at most 1". Do you have any idea @PelicanFive? $\endgroup$ – ilirosmanaj Nov 21 '17 at 20:26

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