  We introduce the concept of weak L-P property for a pair of Banach spaces, in
the study of extreme contractions. We give examples of pairs of Banach spaces
(not) satisfying weak L-P property and apply the concept to compute the exact
number of extreme contractions between a particular pair of polygonal Banach
spaces. We also study the optimality of our results on the newly introduced
weak L-P property for a pair of Banach spaces, by considering appropriate
examples.
