A note on potentially $K_4-e$ graphical sequences

A sequence $S$ is potentially $K_4-e$ graphical if it has a realization containing a $K_4-e$ as a subgraph. Let $\sigma (K_4-e,n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_4-e, n)$ispotentially$K_4-e$ graphical. Gould, Jacobson, Lehel raised the problem of determining the value of $\sigma (K_4-e,n)$. In this paper, we prove that $\sigma (K_4-e,n)=2[(3n-1)\mathrm{/}2]$ for $n\ge 7$, and $n=4,5,$ and $\sigma (K_4-e,6)=20$.