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

TL;DR

This paper investigates the computational complexity of the Minimum Cover Spanning Tree problem across various graph classes, establishing NP-completeness results and providing efficient algorithms for specific subclasses.

Contribution

It establishes the intractability of MCST on certain graph classes and offers fixed-parameter and linear-time algorithms for others, resolving open questions in the field.

Findings

NP-complete for bipartite planar graphs of max degree 4

NP-complete for unit disk graphs

FPT algorithm parameterized by clique-width and linear-time for interval graphs

Abstract

In the context of algorithm theory, various studies have been conducted on spanning trees with desirable properties. In this paper, we consider the \textsc{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 \textsc{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.