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 ≅ K 4 − , where K 4 − stands for the graph obtained from K 4 by deleting one edge. Moreover, we show that either | N G ( V ( H ) ) | = 5 or | N G ( V ( H ) ) | = 6 and around H there is one of two specified structures called a K 4 − -configuration and a split K 4 − -configuration.