{"title":"On Hanf-equivalence and the number of embeddings of small induced subgraphs","authors":"S. Kreutzer, Nicole Schweikardt","doi":"10.1145/2603088.2603148","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":20649,"journal":{"name":"Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"49 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2014-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2603088.2603148","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
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.