稀疏图的幂半图阶的界
Bounds on half graph orders in powers of sparse graphs
摘要：
半图及其变体，如梯子，半梯子和共匹配，是编码图中的全序的组合对象。阿德勒和阿德勒的作品（Eur. J. Comb.; 2014）和FabiaBroski等人（STACS; 2019）证明，在稀疏图的幂中，无法找到任意大的此类对象。然而，这些证明要么是非建设性的，或只提供松散的上界的半图和半梯子的顺序。在这项工作中，我们提供了近紧的渐近下限和上界的最大阶的半图，参数化的距离，在以下类别的稀疏图：可平面图，图有界最大程度，图有界的路径宽度或树宽度，和图不包括一个固定的集团作为一个未成年人。我们的工作中最有意义的部分是可平面图的上界。在这里，我们采用结构图论的技术来分析半梯子可平面图通过笼的概念，揭示了半梯子的拓扑结构。作为这个证明的一个重要组成部分，我们还陈述并证明了一个新的结构性结果，得到了可平面图类中邻域复杂度的完全多项式界。
英文摘要：
Half graphs and their variants, such as ladders, semi-ladders and co-matchings, are combinatorial objects that encode total orders in graphs. Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabiański et al. (STACS; 2019) prove that in the powers of sparse graphs, one cannot find arbitrarily large objects of this kind. However, these proofs either are non-constructive, or provide only loose upper bounds on the orders of half graphs and semi-ladders. In this work we provide nearly tight asymptotic lower and upper bounds on the maximum order of half graphs, parameterized on the distance, in the following classes of sparse graphs: planar graphs, graphs with bounded maximum degree, graphs with bounded pathwidth or treewidth, and graphs excluding a fixed clique as a minor. The most significant part of our work is the upper bound for planar graphs. Here, we employ techniques of structural graph theory to analyze semi-ladders in planar graphs through the notion of cages, which expose a topological structure in semi-ladders. As an essential building block of this proof, we also state and prove a new structural result, yielding a fully polynomial bound on the neighborhood complexity in the class of planar graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
