Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization
containing a $K_{m}-H$ as a subgraph. Let $sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $sigma(S)geq sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $sigma (K_{r+1}-U, n)$ for $ngeq 5r+18, r+1 geq k geq 7,$ $j geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 bigcup P_3$ but not contains a cycle on 4 vertices and not contains $Z_4$. There are a number of graphs on $k$ vertices and $j$ edges which contains a graph $(K_{3} bigcup P_{3})$ but not contains a cycle on 4 vertices and not contains $Z_4$. (for example, $C_3bigcup C_{i_1} bigcup C_{i_2} bigcup >... bigcup C_{i_p}$ $(i_j eq 4, j=2,3,..., p, i_1 geq 5)$, $C_3bigcup P_{i_1} bigcup P_{i_2} bigcup ... bigcup P_{i_p}$ $(i_1 geq 3)$, $C_3bigcup P_{i_1} bigcup C_{i_2} bigcup >... bigcup C_{i_p}$ $(i_j eq 4, j=2,3,..., p, i_1 geq 3)$, etc)
