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
$n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ 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 $Z_4$.