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In this paper, we study three connection games among the most widely played: havannah , twixt , and slither . We show that determining the outcome of an arbitrary input position is PSPACE -complete in all three cases. Our reductions are based on the popular graph problem generalized geography and on hex itself. We also consider the complexity of generalizations of hex parameterized by the length of the solution and establish that while short generalized hex is W[1] -hard, short hex is FPT . Finally, we prove that the ultra-weak solution to the empty starting position in hex cannot be fully adapted to any of these three games. [ABSTRACT FROM AUTHOR]

Abstract: Gale [D. Gale, The game of HEX and the Brouwer fixed-point theorem, American Mathematical Monthly 86 (1979) 818–827] has shown that the so called HEX game theorem that any HEX game has one winner is equivalent to the Brouwer fixed point theorem. In this paper we will show that under some assumptions about marking rules of HEX games, the HEX game theorem for a 6×6 HEX game is equivalent to the Duggan–Schwartz theorem for strategy-proof social choice correspondences [J. Duggan, T. Schwartz, Strategic manipulability without resoluteness or shared beliefs: Gibbard–Satterthwaite generalized, Social Choice and Welfare 17 (2000) 85–93] that there exists no social choice correspondence which satisfies the conditions of strategy-proofness, non-imposition, residual resoluteness, and has no dictator. [Copyright &y& Elsevier]