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

A spanning tree with no vertices of degree from 2 to k of a graph is called a [ 2 , k ] -spanning tree ( [ 2 , k ] -ST) of the graph. In this paper, we consider sets of forbidden subgraphs that imply the existence of a [ 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 4 as an induced subgraph, and characterize such graphs having a [ 2 , 2 ] -ST.