Graph games with an annihilation rule, as introduced by Conway, Fraenkel and Yesha, are studied under the misère play rule for progressively finite graphs that satisfy a condition on the reversibility of non-terminal Sprague-Grundy zeros to Sprague-Grundy ones. Two general theorems on the Sprague-Grundy zeros and ones are given, followed by two theorems characterizing the set of P-positions under certain additional conditions. Application is made to solving many subtraction games, and solutions to two games not covered by the general theory are presented indicating a direction for future research.