Before revision:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
After revision:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, and attain the minimum flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.