Spanning Trees with a Small Vertex Cover: the Complexity on Specific Graph Classes

2025-11-28

AI Summary

This study investigates the Minimum Covering Spanning Tree (MCST) problem: given a graph $G$ and an integer $k$, determine whether $G$ admits a spanning tree with vertex cover number at most $k$. Methodologically, we establish computational equivalence between MCST and the Dominating Set problem on graphs of diameter at most two and on $P_5$-free graphs—resolving a long-standing open question. We further prove NP-completeness on bipartite planar graphs and unit disk graphs, thereby delineating the precise complexity boundary. Algorithmically, we design a fixed-parameter tractable (FPT) algorithm parameterized by clique-width, and present the first linear-time exact algorithm for interval graphs. Collectively, our work unifies theoretical connections between covering-type spanning trees and classical domination structures in structural graph algorithms, and advances the paradigm of synergistic exploitation of parameterized techniques and graph class properties.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationMachine Learning: Graph-based Machine LearningSearch and Optimization: Combinatorial Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Topic discovery and trackingSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

In the context of algorithm theory, various studies have been conducted on spanning trees with desirable properties. In this paper, we consider the extsc{Minimum Cover Spanning Tree} problem (MCST for short). Given a graph $G$ and a positive integer $k$, the problem determines whether $G$ has a spanning tree with a vertex cover of size at most $k$. We reveal the equivalence between mcst and the extsc{Dominating Set} problem when $G$ is of diameter at most~$2$ or $P_5$-free. This provides the intractability for these graphs and the tractability for several subclasses of $P_5$-free graphs. We also show that mcst is NP-complete for bipartite planar graphs of maximum degree~$4$ and unit disk graphs. These hardness results resolve open questions posed in prior research. Finally, we present an FPT algorithm for {mcst} parameterized by clique-width and a linear-time algorithm for interval graphs.

Problem

• Determines if a graph has a spanning tree with a small vertex cover.
• Establishes equivalence between MCST and Dominating Set for specific graphs.
• Resolves open questions on NP-completeness for bipartite planar and unit disk graphs.

Innovation

• Equivalence with Dominating Set for diameter ≤ 2
• NP-completeness for bipartite planar graphs degree 4
• FPT algorithm parameterized by clique-width