寻找HIST：弦性、结构参数与直径
Finding a HIST: Chordality, Structural Parameters, and Diameter
AI总结：
本文针对同胚不可约生成树（HIST）寻找问题，从结构与算法角度开展研究，明确了不同直径弦图的判定复杂度，给出多种参数下的复杂度结论与FPT算法，厘清了该问题的结构与计算边界。
摘要：
同胚不可约生成树（homeomorphically irreducible spanning tree，HIST）是一种不存在2度顶点的生成树，可作为图的结构最简骨干结构。尽管已有大量研究从结构角度探讨HIST的存在性，但寻找HIST的算法复杂度仍未得到充分厘清。本文从结构与算法双视角对HIST问题展开全面研究。我们提出了一项简洁的刻画结论，精准描述了直径不超过3的弦图中哪些存在HIST，由此得到了针对该类图的多项式时间判定流程。与之相对，我们证明该问题在直径为4的强弦图上是NP完全的。从参数化复杂度角度，我们证实HIST问题以团宽度为参数时是W[1]难的，这意味着该问题在一般稠密图中不太可能被高效求解。另一方面，我们提出了分别以树宽、模宽度、簇顶点删除数为参数的固定参数可处理（fixed-parameter tractable，FPT）算法。具体而言，我们开发了以模宽度k为参数、时间复杂度为$O^*(4^{k})$的算法，还基于核化技术提出了以簇顶点删除数为参数的FPT算法，该技术可在保留HIST存在性的前提下限制团的大小。这些结果共同让我们对HIST问题的结构边界与计算边界有了更清晰的认知。
英文摘要：
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.
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