An extremal problem for a graphic sequence to have a realization containing every 2-tree with prescribed size

A graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbors are adjacent, and G − v is a 2-tree. Clearly, if G is a 2-tree on n vertices, then |E(G)| = 2n − 3. A non-increasing sequence π = (d1, . . . , dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795–802) proved that if k ≥ 2, n ≥ 9 2 k + 19 2 k and π = (d1, . . . , dn)