The 2-extendability of 5-connected graphs on the Klein bottle

A graph is said to be k -extendable if any independent set of k edges extends to a perfect matching. In this paper, we shall characterize the forbidden structures for 5-connected graphs on the Klein bottle to be 2-extendable. This fact also gives us a sharp lower bound of representativity of 5-connected graphs embedded on the Klein bottle to have such a property, which was considered in Kawarabayashi et al. (submitted for publication) [4].