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In this paper, we first generalize a theorem for counting the number of facesof 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. Thenwe extend Stahl's bounds for the average genus of the vertex-amalgamation ofgraphs [S. Stahl, Discrete Math. 142 (1995), 235--245] to signed graphs.

The average Euler-genus of the vertex-amalgamation of signed graphs