Abstract: In this paper, we show that for a locally LEW-embedded 3-connected graph G in orientable surface, the following results hold:1) Each of such embeddings is minimum genus embedding;2) The facial cycles are precisely the induced nonseparating cycles which implies the uniqueness of such embeddings;3) Every overlap graph O(G,C) is a bipartite graph and G has only one C-bridge H such that CUH is nonplanar provided C is a contractible cycle shorter than every noncontractible cycle containing an edge of C. This extends the results of C Thomassen’s work on LEW-embedded graphs. 

Key words:  (locally)LEW-embedding; , C-bridge; , overlap , graph