Minimum Spanning Trees with Bounded Degrees of Vertices in a Specified Stable Set

TL;DR

This paper studies a specialized minimum spanning tree problem with degree constraints on a stable set of vertices, providing a polynomial-time solution for the relaxed case using matroid intersection.

Contribution

It offers a new, concise proof of a known characterization and extends the problem to weighted graphs, enabling polynomial-time solutions for minimum-cost trees under certain conditions.

Findings

The problem is NP-complete in general but becomes tractable when restrictions apply to a stable set.

A new proof of Frank's characterization using matroid intersection is provided.

Minimum-cost trees with degree constraints on a stable set can be found efficiently in weighted graphs.

Abstract

Given a graph $G$ and sets $\{\alpha_v~|~v \in V(G)\}$ and $\{\beta_v~|~v \in V(G)\}$ of non-negative integers, it is known that the decision problem whether $G$ contains a spanning tree $T$ such that $\alpha_v \le d_T (v) \le \beta_v $ for all $v \in V(G)$ is $NP$-complete. In this article, we relax the problem by demanding that the degree restrictions apply to vertices $v\in U$ only, where $U$ is a stable set of $G$. In this case, the problem becomes tractable. A. Frank presented a result characterizing the positive instances of that relaxed problem. Using matroid intersection developed by J. Edmonds, we give a new and short proof of Frank's result and show that if $U$ is stable and the edges of $G$ are weighted by arbitrary real numbers, then even a minimum-cost tree $T$ with $\alpha_v \le d_T (v) \le \beta_v $ for all $v \in U$ can be found in polynomial time if such a tree exists.