Forbidden subgraphs restricting vertices of degree two in a spanning tree

For a tree T, let V-2(T) denote the set of vertices of T having degree 2. Let G be a connected graph. A spanning tree T of G with V-2(T) = theta is called a homeomorphically irreducible spanning tree (or a HIST) of G. We focus on two relaxations of HISTs as follows: (1) A spanning tree T of G such that the maximum order of components of the subgraph of T induced by V-2(T) is bounded. (2) A spanning tree T of G such that |V-2(T)|is bounded. A spanning tree satisfying (1) was recently introduced by Lyngsie and Merker, and a spanning tree satisfying (2) is known as a tool for constructing a HIST. In this paper, we define a star-path system, which is a useful concept for finding a spanning tree satisfying (1) or (2) (or both). To demonstrate how the concept works, we characterize forbidden subgraph conditions forcing connected graphs to have such spanning trees.