Finding a HIST: Chordality, Structural Parameters, and Diameter

TL;DR

This work analyzes the problem of finding a HIST, a spanning tree with no degree-2 vertices, from both structural and algorithmic angles. It delivers a polynomial-time characterization and recognition algorithm for chordal graphs with diameter at most 3, and establishes NP-hardness for strongly chordal graphs with diameter 4, delineating sharp complexity boundaries. It also advances the algorithmic toolkit with an exact $4^n$-type dynamic program and several parameterized algorithms, including $O^*(4^{k})$-time modular-width, MSO$_2$-based treewidth, and cluster deletion-number FPT results. Together, these results map out the tractability frontier for HIST across graph classes and structural parameters, informing both theory and practical computation.

Abstract

A homeomorphically irreducible spanning tree (HIST) is a spanning tree with no degree-2 vertices, serving as a structurally minimal backbone of a graph. While the existence of HISTs has been widely studied from a structural perspective, the algorithmic complexity of finding them remains less understood. In this paper, we provide a comprehensive investigation of the HIST problem from both structural and algorithmic viewpoints. We present a simple characterization that precisely describes which chordal graphs of diameter at most~3 admit a HIST, leading to a polynomial-time decision procedure for this class. In contrast, we show that the problem is NP-complete for strongly chordal graphs of diameter~4. From the perspective of parameterized complexity, we establish that the HIST problem is W[1]-hard when parameterized by clique-width, indicating that the problem is unlikely to be efficiently solvable in general dense graphs. On the other hand, we present fixed-parameter tractable (FPT) algorithms when parameterized by treewidth, modular-width, or cluster vertex deletion number. Specifically, we develop an $O^*(4^{k})$-time algorithm parameterized by modular-width~$k$, and an FPT algorithm parameterized by the cluster vertex deletion number based on kernelization techniques that bound clique sizes while preserving the existence of a HIST. These results together provide a clearer understanding of the structural and computational boundaries of the HIST problem.

Figures (1)

• Figure 1: Parameterized complexity of HIST with respect to structural graph parameters. The connection between two parameters means that the upper parameter $p$ is bounded by some computable function $f(\cdot)$ of the lower parameter $q$, i.e., $p \le f (q)$. The double and rounded rectangles indicate that the problem is W[1]-hard and fixed-parameter tractable, respectively.

Theorems & Definitions (25)

• proposition thmcounterproposition
• lemma thmcounterlemma: DBLP:journals/jgt/ShanT23
• theorem thmcountertheorem
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary
• theorem thmcountertheorem
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary