What is this graph problem, and how hard is it?

My problem is quite simple to state, so it surely must have a name:

Given a graph $$G=(V,E)$$ with edge weights $$w(e) \in \mathbb{Z}$$, find a $$V' \subseteq V$$ that maximizes $$\sum_{e \in E' } w(e)$$, where $$E' = \{ (u,v) | u \in V' \vee v\in V'\}$$ is the edge set covered by $$V'$$.

All I can find is variants involving vertex weights, which allows for easier (as far as I can see) reductions from classic problems to prove NP-hardness.

• Related cs.stackexchange.com/questions/84555/… ; you are looking at an induced subgraph with maximal weight. I tried quick googling around this but only found variants with positive weights. That may still be an entry point for you.
– holf
Commented Nov 10, 2022 at 22:16
• Thanks, though I think an induced subgraph requires both ends to be included in the vertex subset. Googling didn't help me either, but stackoverflow suggested a related question immediately, which covers a simple reduction from Independent Set. So I was able to answer my own question. Commented Nov 10, 2022 at 22:48
• @ChandraChekuri, edge weights can be negative, so that might not be the optimal solution.
– D.W.
Commented Nov 11, 2022 at 2:21
• @holf the problem I have in mind requires just one vertex to be selected. Isn't that what I'm saying with $u \in V' \vee v \in V'$? Commented Nov 11, 2022 at 7:41
• Yes my bad, I did not even parse the $\lor$ and was too focused on having both vertices in $V'$.
– holf
Commented Nov 11, 2022 at 7:57