Subgraph induced by the set of degree 5 vertices in a contraction critically 5-connected graph

An edge of a 5-connected graph is said to be contractible if the contraction of the edge results in a 5-connected graph. A 5-connected graph with no contractible edge is said to be contraction critically 5-connected. Let G be a contraction critically 5-connected graph and let H be a component of the subgraph induced by the set of degree 5 vertices of G. Then it is known that |V(H)|≥4. We prove that if |V(H)|=4, then H≅K4−, where K4− stands for the graph obtained from K4 by deleting one edge. Moreover, we show that either |NG(V(H))|=5 or |NG(V(H))|=6 and around H there is one of two specified structures called a K4−-configuration and a split K4−-configuration.