Exact solution to an extremal problem on graphic sequences with a realization containing every $2$-tree on $k$ vertices

De-Yan Zeng, Dong-Yang Zhai, Jian-Hua Yin

A simple graph $G$ is an {\it 2-tree} if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is an 2-tree. Clearly, if $G$ is an 2-tree on $n$ ver…