Forbidden Subgraphs Restricting Vertices of Degree Two in a Spanning Tree

For a tree T, let V₂(T) denote the set of vertices of T having degree $2$. Let G be a connected graph. A spanning tree T of G with V₂(T)=∅ 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 V₂(T) is bounded.(2) A spanning tree T of G such that |V₂(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 an SP-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.