We show a fast algorithm for determining the set of edges in a planar undirected unweighted graph, whose deletion reduces the maximum flow between two fixed vertices. This is a special case of the max flow vitality problem, that has been efficiently solved for general undirected graphs and st-planar graphs. The vitality of an edge of a graph with respect to the maximum flow between two fixed vertices s and t is defined as the reduction of the maximum flow caused by the removal of that edge. In this paper we show that the set of edges having vitality greater than zero in a planar undirected unweighted graph with n vertices can be found in O(n log n) worst-case time and O(n) space.