有序图的饱和

V. Boskovic, Balázs Keszegh
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引用次数: 0

摘要

近年来,$0$-$1$矩阵的饱和问题引起了人们的广泛关注。这个问题可以看作是有序二部图的饱和问题。在此基础上,我们研究了有序图和循环有序图的饱和问题。我们证明二分法在这两种情况下也成立,即对于一个(循环)有序图,它的饱和函数要么是有界的,要么是线性的。我们还确定了(循环)有序图的大类别的数量级,给出了展示两种可能行为的无限多个示例,回答了P\'alv\ ' olgyi问题。特别地,在有序情况下,我们定义了有序匹配的一个自然子类——链接匹配类,并对它们进行了系统的研究,重点研究了最多只有三个链接的链接匹配,并证明了其中许多匹配具有有界饱和函数。在有序和循环有序的情况下,我们还考虑了半饱和问题,其中二分法也成立,我们甚至可以完全表征具有有界半饱和函数的图。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Saturation of Ordered Graphs
Recently, the saturation problem of $0$-$1$ matrices gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study of the saturation problem of ordered and cyclically ordered graphs. We prove that dichotomy holds also in these two cases, i.e., for a (cyclically) ordered graph its saturation function is either bounded or linear. We also determine the order of magnitude for large classes of (cyclically) ordered graphs, giving infinite many examples exhibiting both possible behaviours, answering a problem of P\'alv\"olgyi. In particular, in the ordered case we define a natural subclass of ordered matchings, the class of linked matchings, and we start their systematic study, concentrating on linked matchings with at most three links and prove that many of them have bounded saturation function. In both the ordered and cyclically ordered case we also consider the semisaturation problem, where dichotomy holds as well and we can even fully characterize the graphs that have bounded semisaturation function.
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