Flexibility of Embeddings of a Halin Graph on the Projective Plane

A basic problem in graph embedding theory is to determine distinct embeddings of planar graphs on higher surfaces. Tutte’s work on graph connectivity shows that wheels or wheel-like configurations plays a key role in 3-connected graphs. In this paper we investigate the flexibility of a Halin grap on N1, the projective plane, and show that embeddings of a Halin graph on N1 is determined by making either a twist or a 3-patchment of a vertex in a wheel. Further more, as applications, we give a correspondence between a Halin graph and its embeddings on the projective plane. Based on this, the numbers of some types of such embeddings are determined