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

TL;DR

The paper tackles a relaxed Minimum Bounded Degree Spanning Tree problem where degree constraints are imposed only on a stable set $U$. It reframes the search for a feasible spanning tree as a matroid intersection problem between a generalized partition matroid $M_1$ (capturing $\alpha_v,\beta_v$ on $\delta_G(v)$ for $v\in U$) and the cycle matroid $M_2$, linking feasible trees to common independent sets of size $|V(G)|-1$. A concise, matroid-based proof of Frank's characterization is provided, and the Edmonds intersection framework yields a polynomial-time algorithm to decide feasibility and, when feasible, compute a minimum-cost tree with $\alpha_v \le d_T(v) \le \beta_v$ for all $v\in U$. The approach integrates rank-function analysis and well-definedness conditions to deliver a practical, theory-backed solution for a broader class of degree-bounded spanning trees. This has potential implications for network design where degree constraints are localized to independent subnetworks.

Abstract

Given a graph $G$ and sets $\{α_v~|~v \in V(G)\}$ and $\{β_v~|~v \in V(G)\}$ of non-negative integers, it is known that the decision problem whether $G$ contains a spanning tree $T$ such that $α_v \le d_T (v) \le β_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 $α_v \le d_T (v) \le β_v $ for all $v \in U$ can be found in polynomial time if such a tree exists.

Theorems & Definitions (16)

• Theorem 1: A. Frank, book, Theorem 9.1.16
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 1
• Lemma 1
• proof
• Proposition 2
• proof : Proof of the Only-If-Part of Theorem \ref{['1']}:
• proof : Proof of the If-Part of Theorem \ref{['1']}: