Abstract: Combined with the edge-connectivity, this paper investigated the upper embeddable graphs with specific minimum degree-sum of vertices in its independent-set, and obtained the following results. (1) Let G be a 2-edge-connected simple graph, if G satisfies the following conditions: for any 3-independent set I in G, ∨x_i ,x_j ∈I (i,j = 1,2,3), d(x_i ,x_j)≧3 (1 ≦ i ≠ j ≦ 3) =>
∑_{i = 1}³ d(x_i) ≧ v + 1(v = | V(G)|}), then G is upper embeddable; (2) Let G be
3-edge-connected simple graph, if G satisfies the following conditions: for any 6-independent set I in G, ∨x_i ,x_j ∈I (1≦i,j≦6), d(x_i,x_j) ≧3(1 ≦ i ≠ j ≦ 6) => ∑_{i = 1}⁶ d(x_i) ≧ v + 1(v = | V(G)|), then G is upper embeddable.

Key words: maximum genus, Betti deficiency, upper embeddable, k-independentset, Graph, maximum genus, Betti deficiency, upper embeddable, k-independentset