  In this paper, we first generalize a theorem for counting the number of faces
of an oriented embedding of a graph that passing through a given cut-edge set
[S. Stahl, Trans. Amer. Math. Soc. 259 (1980), 129--145] to all surfaces. Then
we extend Stahl's bounds for the average genus of the vertex-amalgamation of
graphs [S. Stahl, Discrete Math. 142 (1995), 235--245] to signed graphs.
