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引用次数: 0
摘要
Nešetřil和Ossona de Mendez最近提出了一个图收敛的新定义,叫做结构收敛。结构收敛框架是基于一阶公式的固定片段满足逻辑公式的概率。选择片段的灵活性允许将稀疏图和密集图的经典收敛概念统一起来。由于该领域相对较年轻,收敛序列的例子范围有限,只有几种构造方法是已知的。我们的目标是通过考虑小工具结构来扩展结构的多样性。我们证明,当限定在句子集合上时,在初等收敛序列上应用小集构造得到一个初等收敛序列。另一方面,我们展示了反例,证明了在没有额外假设的情况下,不可能推广到完全一阶收敛。给出了保证完全收敛的几个充分条件。其中之一指出,如果替换的边在原始结构序列中是密集的,则所得序列是一阶收敛的。
Nešetřil and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed fragment of first-order formulas. The flexibility of choosing the fragment allows to unify the classical notions of convergence for sparse and dense graphs. Since the field is relatively young, the range of examples of convergent sequences is limited and only a few methods of construction are known. Our aim is to extend the variety of constructions by considering the gadget construction. We show that, when restricting to the set of sentences, the application of gadget construction on elementarily convergent sequences yields an elementarily convergent sequence. On the other hand, we show counterexamples witnessing that a generalization to the full first-order convergence is not possible without additional assumptions. We give several different sufficient conditions to ensure the full convergence. One of them states that the resulting sequence is first-order convergent if the replaced edges are dense in the original sequence of structures.
期刊介绍:
COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are
- Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups).
- Combinatorial optimization.
- Combinatorial aspects of geometry and number theory.
- Algorithms in combinatorics and related fields.
- Computational complexity theory.
- Randomization and explicit construction in combinatorics and algorithms.