An ascending subgraph decomposition of a graph $G$ is a decomposition of $G$ to subgraphs $G_1,G_2,\dots ,G_n$ of $G$ such that $\binom{n+1}{2} \leq |E(G)|\leq\binom{n+2}{2},\ G=\bigcup_{i=1}^{n} G_{i},\ G_{i}\subset G_{i+1}$ for $i=1,2,\dots,n-1$, and $G_i$ have no isolated vertex. Y. Alavi\ et al.\ [Congr. Numer. {\bf 58} (1987), 7--14; MR0944683 (89d:05136)] introduced the concept of ascending subgraph decompositions of graphs and conjectured that any graph has an ascending subgraph decomposition. Hence it was proved that $K_n -H_i$ has an ascending subgraph decomposition for $n-1\leq i\leq 2n$, where $K_n$ is the complete graph with $n$ vertices and $H_i$ is a subgraph of $K_n$ with $i$ vertices [R. J. Faudree, A. Gyárfás\ and R. H. Schelp, Congr. Numer. {\bf 59} (1987), 49--54; MR0944946 (89d:05142); K. J. Ma\ and H. T. Chen, Adv. in Math. (China) {\bf 26} (1997), no.~1, 66--71; MR1457610 (98b:05082)]. In this paper, the authors prove that $K_n -H_i$ has an ascending subgraph decomposition to $K_{1,1},\ K_{1,2}, \dots,K_{1,n-5},G_{n-4}$ for $i=2n+1,\ n\geq 8$ and $i=2n+2,\ n\geq 10$.