Let $K_k$, $C_k$, $T_k$, and $P_{k}$ denote a complete graph on $k$
vertices, a cycle on $k$ vertices, a tree on $k+1$ vertices, and a
path on $k+1$ vertices, respectively. 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 $K_{m}$). 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
$\sigma (K_{r+1}-H, n)$ for
$n\geq 4r+10, r\geq 3, r+1 \geq k \geq 4$ where $H$ is a graph on $k$
vertices which
contains a tree on $4$ vertices but
not contains a cycle on $3$ vertices. We also determine the values of
$\sigma (K_{r+1}-P_2, n)$ for
$n\geq 4r+8, r\geq 3$.