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

TL;DR

The paper investigates MCST, the problem of finding a spanning tree with a small vertex cover. It establishes a tight connection between MCST and Dominating Set on diameter-2 and P5-free graphs, yielding equality of the two central parameters in these classes and matching algorithmic consequences. It also proves NP-hardness for planar bipartite graphs with maximum degree 4 and for unit disk graphs, while delivering an FPT algorithm by clique-width via an MSO1 reformulation and a linear-time solution for interval graphs. Together, these results map the complexity landscape of MCST across diverse graph classes and provide practical algorithms for important subclasses. The work resolves open questions and highlights a nuanced boundary between tractability and hardness for MCST.

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.

Figures (8)

• Figure 1: (a) A Complete graph $G$, (b) a spanning tree of $G$ with a vertex cover of size $1$, and (c) a spanning tree of $G$ with a vertex cover of size $3$. The spanning trees are shown in bold green lines, and the corresponding vertex covers are indicated by black circles.
• Figure 2: Known and our results with respect to graph classes. Each arrow $A \rightarrow B$ represents that the graph class $B$ is a subclass of the graph class $A$.
• Figure 3: (a) An example of illustrating a rectilinear representation of a formula $\varphi = (x_1 \lor x_3 \lor x_4) \land (x_1 \lor x_2 \lor x_3) \lor (\overline{x_3} \lor \overline{x_4}) \land (\overline{x_1} \lor \overline{x_2} \lor \overline{x_4})$ and (b) the graph $G$ constructed from $\varphi$.
• Figure 4: (a) The variable gadget $U_i$. If the variable $x_i$ is assigned True (resp. False), then the corresponding tree is depicted in green and its vertex cover is indicated by black circles in (b) (resp. (c)).
• Figure 5: A Connector gadget between variable gadgets. There are two ways to connect the connector gadget depending on the degree of $u_{i+1}$.

Theorems & Definitions (37)

• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof
• remark thmcounterremark
• corollary thmcountercorollary
• corollary thmcountercorollary
• corollary thmcountercorollary
• theorem thmcountertheorem
• lemma thmcounterlemma: $\ast$