A generalization of the rectangle condition

In this paper, we define the connecting rectangle condition to check whether or not a Heegaard splitting is strongly irreducible which is a variation of the rectangle condition by Casson and Gordon. We define a general version of the rectangle condition. Moreover, with a similar condition defined on an n-bridge decomposition, we can check whether or not the Hempel distance of an n-bridge decomposition is greater than or equal to two.