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arXiv · 2511.22912

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

Toranosuke Kokai·Akira Suzuki·Takahiro Suzuki·Yuma Tamura·Xiao Zhou
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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.

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Toranosuke Kokai, Akira Suzuki, Takahiro Suzuki, Yuma Tamura, Xiao Zhou. 2025-11-28. Spanning Trees with a Small Vertex Cover: the Complexity on Specific Graph Classes. https://arxiv.org/abs/2511.22912

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