  The paper introduces two player connectivity games played on finite bipartite
graphs. Algorithms that solve these connectivity games can be used as
subroutines for solving M\"uller games. M\"uller games constitute a well
established class of games in model checking and verification. In connectivity
games, the objective of one of the players is to visit every node of the game
graph infinitely often. The first contribution of this paper is our proof that
solving connectivity games can be reduced to the incremental strongly connected
component maintenance (ISCCM) problem, an important problem in graph algorithms
and data structures. The second contribution is that we non-trivially adapt two
known algorithms for the ISCCM problem to provide two efficient algorithms that
solve the connectivity games problem. Finally, based on the techniques
developed, we recast Horn's polynomial time algorithm that solves explicitly
given M\"uller games and provide an alternative proof of its correctness. Our
algorithms are more efficient than that of Horn's algorithm. Our solution for
connectivity games is used as a subroutine in the algorithm.
