小诱导子图的半等价和嵌入数

S. Kreutzer, Nicole Schweikardt
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引用次数: 2

摘要

两个图对于半径r是半等价的,如果在它们的顶点集之间有一个双射,它保留了半径r的顶点邻域的同构类型。对于r = 1,这意味着两个图具有相同的度序列。本文将半等价与图论的子图等价概念联系起来。为了使这个概念适用于不一定连通的图,我们首先以一种合适的方式将连通图半径的概念推广到一般图,我们称之为广义半径。如果对于半径为r的所有图S,在G和H中与S同构的诱导子图的数目相等,则说明两个图G和H在半径为r的范围内是子图等价的。我们证明了半径为r的半等价等价于半径为r的范围内的子图等价,从而将纯逻辑概念和图论概念强有力地联系起来。据此定义了s阶以下的子图等价的概念,其中考虑了所有s阶以下的图。作为一个推论,我们得到关于半径r的半等价意味着s阶的子图等价,只要r≥3s/4。特别地,这意味着两个半径为3 /4的半等价图满足最多为s的量词的合取查询的完全相同的并。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On Hanf-equivalence and the number of embeddings of small induced subgraphs
Two graphs are Hanf-equivalent with respect to radius r if there is a bijection between their vertex sets which preserves the isomorphism types of the vertices' neighbourhoods of radius r. For r = 1 this means that the graphs have the same degree sequence. In this paper we relate Hanf-equivalence to the graph-theoretical concept of subgraph equivalence. To make this concept applicable to graphs that are not necessarily connected, we first generalise the notion of the radius of a connected graph to general graphs in a suitable way, which we call the generalised radius. We say that two graphs G and H are subgraph-equivalent up to generalised radius r if for all graphs S of generalised radius r, the number of induced subgraphs isomorphic to S is the same in G and H. We prove that Hanf-equivalence with respect to radius r is equivalent to subgraph-equivalence up to generalised radius r, thereby relating the purely logical and the graph-theoretical concepts in a very strong way. The notion of subgraph-equivalence up to order s is defined accordingly, where all graphs S of order at most s are taken into account. As a corollary we obtain that Hanf-equivalence with respect to radius r implies subgraph-equivalence up to order s, provided that r ≥ 3s/4. In particular, this implies that two graphs which are Hanf-equivalent with respect to radius 3s/4 satisfy exactly the same unions of conjunctive queries of quantifier rank at most s.
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