Forbidden Subgraphs Restricting Vertices of Degree Two in a Spanning Tree
Abstract
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.