Forbidden subgraphs and the existence of a spanning tree without small degree stems

A spanning tree with no vertices of degree from 2 to kk of a graph is called a [2,k][2,k]-spanning tree   ([2,k][2,k]-ST  ) of the graph. In this paper, we consider sets of forbidden subgraphs that imply the existence of a [2,k][2,k]-ST in a connected graph of sufficiently large order, and give the characterization of such sets. We also focus on connected graphs that contain no path of order 44 as an induced subgraph, and characterize such graphs having a [2,2][2,2]-ST.