  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 $\sigma (K_{r+1}-Z, n)$ for $n\geq
5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices and
$j$ edges which contains a graph $Z_4$ but not contains a cycle on 4 vertices.
We also determine the values of $\sigma (K_{r+1}-Z_4, n)$, $\sigma
(K_{r+1}-(K_4-e), n)$, $\sigma (K_{r+1}-K_4, n)$ for $n\geq 5r+16, r\geq 4$.
There are a number of graphs on $k$ vertices and $j$ edges which contains a
graph $Z_4$ but not contains a cycle on 4 vertices.
