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I have 2 different but similar problems, one belongs to NP and one to L and I don't understand why.

First problem: Input: an undirected graph G with n^2 vertices. Question: Is there exist in G a matching of size n AND an independent set of size n?

Second problem: Input: an undirected graph G with n^2 vertices. Question: Is there exist in G a matching of size n OR an independent set of size n?

The AND problem is in NP and the OR problem is in L. Can someone explain why? Thanks.

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    – D.W. ♦
    Jan 23, 2015 at 17:17

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Note that both problems are in NP. It's just that the first is also NP-hard, and the second is also in L.

Hint for the first problem: Prove that it's NP-hard by reduction from independent set. Given a graph $G=(V,E)$ and a parameter $k \leq |V|$, add an independent set of size $|V|-k$ and a clique of size $|V|^2-|V| \geq 2|V|$ (the case $|V| < 3$ has to be handled separately), and connect all vertices of the clique to all other vertices.

Hint for the second problem: Suppose that $G$ doesn't have an independent set of size $n$. Show that it has a matching of size $n$ by dividing the $n^2$ vertices into $n$ sets of $n$ vertices, and picking an edge from each.

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