This paper continues the research on the computational aspects of Halpern and Pearl's causes and explanations in the structural model approach. To this end, we first explore how an instance of deciding weak cause can be reduced to an equivalent instance in which irrelevant variables in the (potential) weak cause and the causal model are removed, which extends previous work by Hopkins. We then present a new characterization of weak cause for a certain class of causal models in which the cau...