The winner of the unbiased triangle game can be decided in O(n^7) time on general graphs, O(n^{ω+1}) on connected K4-containing graphs, and O(n^3) when the edge-triangle incidence graph is a cactus.
OntheChvátal-ErdösTriangleGame.Electron.J.Comb.,18(1),2011
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Faster Algorithms for Deciding the Unbiased Maker-Breaker Triangle Game on General Graphs
The winner of the unbiased triangle game can be decided in O(n^7) time on general graphs, O(n^{ω+1}) on connected K4-containing graphs, and O(n^3) when the edge-triangle incidence graph is a cactus.