Hilbert 片段

中文摘要 AI 辅助

在先前的工作中，我们证明了区间序的长度多面体具有唯一的 Hilbert 基，且该基的每个元素都是二元向量。这些二元向量的支撑集构成一个集系统，其交集图是共可比较图。这自然引出了一个问题：哪些图可以作为此类 Hilbert 基支撑集的交集图。我们引入了 Hilbert 片段，定义为这些交集图的诱导子图。因此，Hilbert 片段构成共可比较图的一个遗传类，记录了区间序长度多面体的 Hilbert 基元素的支撑集之间的交集模式。我们通过一个公理化定义的顶点标记（称为轨道标记）来刻画 Hilbert 片段，并给出应用实例，说明即使 Hilbert 片段本身不是区间图，这种刻画也能揭示其潜在的一维结构。

英文摘要

In earlier work, we proved that the length polyhedron of an interval order has a unique Hilbert basis and that every element of this basis is a binary vector. The supports of these binary vectors form a set system whose intersection graph is a cocomparability graph. This leads naturally to the question of which graphs can arise as intersection graphs of such Hilbert-basis supports. We introduce Hilbert fragments, defined as induced subgraphs of these intersection graphs. Thus Hilbert fragments form a hereditary class of cocomparability graphs recording the intersection patterns among the supports of Hilbert-basis elements of interval-order length polyhedra. We characterize Hilbert fragments by an axiomatically defined vertex-labeling, called a track-labeling, and give applications illustrating how this characterization can reveal an underlying one-dimensional structure even when the Hilbert fragments themselves are not interval graphs.

发表机构

• University of Louisville（路易斯维尔大学）
• Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）

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