Abstract: Let G be a connected graph with ξ(G)=k≥1.If ξ(G\e)=k-1,G is called to be a (ξ,1)-critical graph.This paper gives the upper embeddability of the ξ-1-critical graphs,and shows that extension of a vertex and extension of a cycle dose not change the ξ-1-cirtical graphs.The new upper embeddable graphs are obtained,enriching the kind and seeking methods.

Key words: Betti deficiency number, maximum genus, upper embeddability, (&xi, ,1)-critical graphs