{"title":"Properties of graphs specified by a regular language","authors":"Volker Diekert, Henning Fernau, Petra Wolf","doi":"10.1007/s00236-022-00427-z","DOIUrl":null,"url":null,"abstract":"<div><p>Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property <span>\\(\\varPhi \\)</span>. What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying <span>\\(\\varPhi \\)</span> in the family? We approach this question by using formal languages for specifying families of graphs, in particular by regular sets of words. We show that certain graph properties can be decided by studying the syntactic monoid of the specification language <i>L</i> if a certain torsion condition is satisfied. This condition holds trivially if <i>L</i> is regular. More specifically, we use a natural binary encoding of finite graphs over a binary alphabet <span>\\(\\varSigma \\)</span>, and we define a regular set <span>\\(\\mathbb {G}\\subseteq \\varSigma ^*\\)</span> such that every nonempty word <span>\\(w\\in \\mathbb {G}\\)</span> defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over <span>\\(\\varSigma \\)</span>. Then, we ask whether the automaton <span>\\(\\mathcal {A}\\)</span> specifies some graph satisfying a certain property <span>\\(\\varPhi \\)</span>. Our structural results show that we can answer this question for all “typical” graph properties. In order to show our results, we split <i>L</i> into a finite union of subsets and every subset of this union defines in a natural way a single finite graph <i>F</i> where some edges and vertices are marked. The marked graph in turn defines an infinite graph <span>\\(F^\\infty \\)</span> and therefore the family of finite subgraphs of <span>\\(F^\\infty \\)</span> where <i>F</i> appears as an induced subgraph. This yields a geometric description of all graphs specified by <i>L</i> based on splitting <i>L</i> into finitely many pieces; then using the notion of graph retraction, we obtain an easily understandable description of the graphs in each piece.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00236-022-00427-z.pdf","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Informatica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00236-022-00427-z","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
引用次数: 3
Abstract
Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property \(\varPhi \). What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying \(\varPhi \) in the family? We approach this question by using formal languages for specifying families of graphs, in particular by regular sets of words. We show that certain graph properties can be decided by studying the syntactic monoid of the specification language L if a certain torsion condition is satisfied. This condition holds trivially if L is regular. More specifically, we use a natural binary encoding of finite graphs over a binary alphabet \(\varSigma \), and we define a regular set \(\mathbb {G}\subseteq \varSigma ^*\) such that every nonempty word \(w\in \mathbb {G}\) defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over \(\varSigma \). Then, we ask whether the automaton \(\mathcal {A}\) specifies some graph satisfying a certain property \(\varPhi \). Our structural results show that we can answer this question for all “typical” graph properties. In order to show our results, we split L into a finite union of subsets and every subset of this union defines in a natural way a single finite graph F where some edges and vertices are marked. The marked graph in turn defines an infinite graph \(F^\infty \) and therefore the family of finite subgraphs of \(F^\infty \) where F appears as an induced subgraph. This yields a geometric description of all graphs specified by L based on splitting L into finitely many pieces; then using the notion of graph retraction, we obtain an easily understandable description of the graphs in each piece.
期刊介绍:
Acta Informatica provides international dissemination of articles on formal methods for the design and analysis of programs, computing systems and information structures, as well as related fields of Theoretical Computer Science such as Automata Theory, Logic in Computer Science, and Algorithmics.
Topics of interest include:
• semantics of programming languages
• models and modeling languages for concurrent, distributed, reactive and mobile systems
• models and modeling languages for timed, hybrid and probabilistic systems
• specification, program analysis and verification
• model checking and theorem proving
• modal, temporal, first- and higher-order logics, and their variants
• constraint logic, SAT/SMT-solving techniques
• theoretical aspects of databases, semi-structured data and finite model theory
• theoretical aspects of artificial intelligence, knowledge representation, description logic
• automata theory, formal languages, term and graph rewriting
• game-based models, synthesis
• type theory, typed calculi
• algebraic, coalgebraic and categorical methods
• formal aspects of performance, dependability and reliability analysis
• foundations of information and network security
• parallel, distributed and randomized algorithms
• design and analysis of algorithms
• foundations of network and communication protocols.