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Assume we have an undirected Graph G = (V,E) and we construct a new Graph G' where two nodes are adjacent if they have a common neighbor node in G. Can someone explain why the following statements are true if we have such a construction G'?

If G has an independent set of size n, then G' has a matching of size n. If G' has an matching of size n, then G has an independent set of size n.

Unfortunately I don't have an idea for this problem

Marc
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  • "...where two nodes are adjacent if they have a common node in G", what does that mean? Do you mean "if they have a common neighbor in G"? – SaiBot Jan 08 '19 at 14:58
  • Oh yes I mean neighbor – Marc Jan 08 '19 at 17:08
  • Do you have some other assumption about the graph? Maybe something about connectivity? Otherwise, consider a graph G with n nodes and no edges. The set of all nodes in G is an independent set and clearly there are no edges in G', so also no matching of size > 0. – zohar.kom Jan 15 '19 at 16:12

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