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引用次数: 0
摘要
阿万(Awan)和贝尔纳迪(Bernardi)提出的 B 多项式和准对称 B 函数将广泛研究的图特多项式和图特对称函数扩展到了数图。在本文中,我们将讨论有关这些数图不变式的一个基本问题,即如何确定由这些不变式唯一表征的数图类别。我们解决了最初由 Awan 和 Bernardi 提出的一个未决问题,即如何识别用一对相对的弧替换图中每一条边所产生的数图。此外,我们还解决了一个更具挑战性的问题,即如何利用准对称函数重构数图。特别是,我们证明了准对称 B 函数可以重建适当毛毛虫的部分对称方向。因此,我们确定路径和非对称正交毛虫的所有方向都可以通过它们的准对称 B 函数来重建。这些结果增加了可通过类对称函数区分的定向树的数量。
Distinguishing and Reconstructing Directed Graphs by their $$\pmb {B}$$ -Polynomials
The B-polynomial and quasisymmetric B-function, introduced by Awan and Bernardi, extends the widely studied Tutte polynomial and Tutte symmetric function to digraphs. In this article, we address one of the fundamental questions concerning these digraph invariants, which is, the determination of the classes of digraphs uniquely characterized by them. We solve an open question originally posed by Awan and Bernardi, regarding the identification of digraphs that result from replacing every edge of a graph with a pair of opposite arcs. Further, we address the more challenging problem of reconstructing digraphs using their quasisymmetric functions. In particular, we show that the quasisymmetric B-function reconstructs partially symmetric orientations of proper caterpillars. As a consequence, we establish that all orientations of paths and asymmetric proper caterpillars can be reconstructed from their quasisymmetric B-functions. These results enhance the pool of oriented trees distinguishable through quasisymmetric functions.