Given a tree on $$$n$$$ nodes, perform the following operation as many times as you like (including zero times): Choose four nodes connected in a chain, and contract them to a single node, merging their adjacency lists. Output the minimum size you can reduce the tree to.
Does anyone know how to find a polynomial time solution for this, or prove that the problem is NP-Complete? It seems very difficult for me. Has anyone seen this problem, or something equivalent, before?
I tried it for very long, and then I gave it to several IGMs and they couldn't solve it, and ChatGPT 5.5 Pro can neither solve it nor prove NP-completeness.
bro we should've tried harder and used it as div 1 F :wilted_rose:
Auto comment: topic has been updated by TAhmed33 (previous revision, new revision, compare).
Elegia could you please help him
I sent this to Fable 5 Max, let's see what happens. He's been thinking for 15 minutes. Unfortunately it looks like he stopped thinking for some reason but hopefully will continue.
Update: no response was generated but my usage was consumed.
Shockingly, after back and forth with 5.5 Pro and Fable, and about 20 bucks in credits, and verifying up to $$$n \lesssim 20$$$, and forcing the models to double-check their work, I got a solution.
Let $$$T$$$ be a tree on $$$n$$$ vertices and let $$$\mu = \mu(T)$$$ denote the size of a maximum matching of $$$T$$$. The minimum size the tree can be reduced to equals
Full proof as image or paste, for some reason Codeforces prevents me from posting it.