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Two players try to keep graph disconnected

Source: IMOC 2021 C6

August 11, 2021
combinatoricsgraph theoryIMOC

Problem Statement

Two people play a game on a graph with 20222022 points. Initially, there are no edges in the graph. They take turns and connect two non-neighbouring vertices with an edge. Whoever makes the graph connected loses. Which player has a winning strategy?
ST, danny2915