Some structural properties of minimally contraction-critically 5-connected graphs

An edge of a kk-connected graph is said to be kk-removable (resp. kk-contractible) if the removal (resp. the contraction ) of the edge results in a kk-connected graph. A kk-connected graph with neither kk-removable edge nor kk-contractible edge is said to be minimally contraction-critically kk-connected. We show that around an edge whose both end vertices have degree greater than 5 of a minimally contraction-critically 5-connected graph, there exists one of two specified configurations. Using this fact, we prove that each minimally contraction-critically 5-connected graph on nn vertices has at least 23n vertices of degree 5.