Summary: The domination number of cylinder graphs, denoted by \(\gamma(P_m\square C_n)\), has previously been determined for \(m<5\) and any \(n\). Here we present an algorithm to determine a formula for \(\gamma(P_m\square C_n)\), when either \(m\) or \(n\) is fixed, in terms of the other parameter. The results include formulae for \(\gamma(P_m\square C_n)\), with \(m\leq 22\) or \(n\leq 30\) as fixed parameters.