Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finding a Minimum Spanning Tree with a Small Non-Terminal Set

Tesshu Hanaka, Yasuaki Kobayashi

TL;DR

The paper studies Minimum Weight Non-Terminal Spanning Tree (MWNST), where every vertex in a designated non-terminal set $V_{ m NT}$ must be internal in a spanning tree, and analyzes both the weighted and unweighted variants. It develops a robust set of parameterized algorithms and kernelizations: a $3k$-vertex kernel for Non-Terminal Spanning Tree, linear kernels for vertex cover and quadratic kernels for max leaf number, an $O^*(2^k)$-time algorithm for MWNST with polynomially bounded integral weights, and an $O^*(2^{\ell})$-time algorithm for MWNST with arbitrary weights, with $k=|V_{ m NT}|$ and $\ell=|E(G[V_{ m NT}])|$. The results also show MSO$_2$ definability of the property, giving FPT by treewidth, and establish hardness fronts including NP-hardness, W[1]-hardness by clique-width, and ETH-based lower bounds. These contributions advance understanding of how structural parameters influence exact solvability for spanning-tree variants and enable efficient algorithms on graphs with small non-terminal sets or restricted subgraph structure.

Abstract

In this paper, we study the problem of finding a minimum weight spanning tree that contains each vertex in a given subset $V_{\rm NT}$ of vertices as an internal vertex. This problem, called Minimum Weight Non-Terminal Spanning Tree, includes $s$-$t$ Hamiltonian Path as a special case, and hence it is NP-hard. In this paper, we first observe that Non-Terminal Spanning Tree, the unweighted counterpart of Minimum Weight Non-Terminal Spanning Tree, is already NP-hard on some special graph classes. Moreover, it is W[1]-hard when parameterized by clique-width. In contrast, we give a $3k$-vertex kernel and $O^*(2^k)$-time algorithm, where $k$ is the size of non-terminal set $V_{\rm NT}$. The latter algorithm can be extended to Minimum Weight Non-Terminal Spanning Tree with the restriction that each edge has a polynomially bounded integral weight. We also show that Minimum Weight Non-Terminal Spanning Tree is fixed-parameter tractable parameterized by the number of edges in the subgraph induced by the non-terminal set $V_{\rm NT}$, extending the fixed-parameter tractability of Minimum Weight Non-Terminal Spanning Tree to the general case. Finally, we give several results for structural parameterization.

Finding a Minimum Spanning Tree with a Small Non-Terminal Set

TL;DR

The paper studies Minimum Weight Non-Terminal Spanning Tree (MWNST), where every vertex in a designated non-terminal set must be internal in a spanning tree, and analyzes both the weighted and unweighted variants. It develops a robust set of parameterized algorithms and kernelizations: a -vertex kernel for Non-Terminal Spanning Tree, linear kernels for vertex cover and quadratic kernels for max leaf number, an -time algorithm for MWNST with polynomially bounded integral weights, and an -time algorithm for MWNST with arbitrary weights, with and . The results also show MSO definability of the property, giving FPT by treewidth, and establish hardness fronts including NP-hardness, W[1]-hardness by clique-width, and ETH-based lower bounds. These contributions advance understanding of how structural parameters influence exact solvability for spanning-tree variants and enable efficient algorithms on graphs with small non-terminal sets or restricted subgraph structure.

Abstract

In this paper, we study the problem of finding a minimum weight spanning tree that contains each vertex in a given subset of vertices as an internal vertex. This problem, called Minimum Weight Non-Terminal Spanning Tree, includes - Hamiltonian Path as a special case, and hence it is NP-hard. In this paper, we first observe that Non-Terminal Spanning Tree, the unweighted counterpart of Minimum Weight Non-Terminal Spanning Tree, is already NP-hard on some special graph classes. Moreover, it is W[1]-hard when parameterized by clique-width. In contrast, we give a -vertex kernel and -time algorithm, where is the size of non-terminal set . The latter algorithm can be extended to Minimum Weight Non-Terminal Spanning Tree with the restriction that each edge has a polynomially bounded integral weight. We also show that Minimum Weight Non-Terminal Spanning Tree is fixed-parameter tractable parameterized by the number of edges in the subgraph induced by the non-terminal set , extending the fixed-parameter tractability of Minimum Weight Non-Terminal Spanning Tree to the general case. Finally, we give several results for structural parameterization.
Paper Structure (15 sections, 26 theorems, 2 equations, 1 figure)

This paper contains 15 sections, 26 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related work
  4. Preliminaries
  5. Kernelization
  6. Linear kernel for $k$
  7. Linear kernel for vertex cover number
  8. Quadratic kernel for max leaf number
  9. Kernel lower bound for vertex integrity
  10. Fixed-Parameter Algorithms
  11. Parameterization by $k$
  12. Parameterization by $\ell$
  13. Parameterization by tree-width
  14. Hardness results
  15. Conclusion

Key Result

Proposition 2

Let $G$ be a connected graph and $F$ be a forest in $G$. Then there is a spanning tree that contains all edges in $F$.

Figures (1)

  • Figure 1: The construction of $\hat{G}$. The shaded areas indicate the vertices of $V_{\rm NT}$ and $\hat{V}_{\rm NT}$. The bold lines are edges in a $2$-expansion of $X$ into $Y$.

Theorems & Definitions (31)

  • Definition 1
  • Proposition 2
  • Proposition 3: GraphParamFLMMRS2009:maxleafGimaHKKO22
  • Proposition 4
  • Theorem 5: Frank1981
  • Lemma 6
  • Lemma 7
  • Corollary 8
  • Definition 9
  • Lemma 10: Expansion lemma Fomin_Hitting_2016Thomasse10
  • ...and 21 more