Properties of Graphs Specified by a Regular Language

Diekert, Volker; Fernau, Henning; Wolf, Petra

doi:10.1007/978-3-030-81508-0_10

Cited by 3 publications

(5 citation statements)

References 11 publications

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“…Finally, between any two occurrences of the factor aaa in w, there must be a factor baab 11. Corollary 4 corrects a misprint in[8, Cor. 3].…”

mentioning

confidence: 82%

“…such that it holds that F α = L when forgetting the annotation. 8 For the verification whether (V , E, ν) ∈ L we can use words in A * B * which are of bounded length and a guess-and-check algorithm. For each (V , E) ∈ F α , we guess its annotation ν.…”

Section: Marked Graphsmentioning

confidence: 99%

“…We thank Dietrich Kuske for pointing out that the formulation of [8, Thm. 3] is not correct as published in the proceedings.…”

Section: Acknowledgementsmentioning

confidence: 99%

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Properties of graphs specified by a regular language

2022

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Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property $$\varPhi $$ Φ . 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 $$\varPhi $$ Φ 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 $$\varSigma $$ Σ , and we define a regular set $$\mathbb {G}\subseteq \varSigma ^*$$ G ⊆ Σ ∗ such that every nonempty word $$w\in \mathbb {G}$$ w ∈ G defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over $$\varSigma $$ Σ . Then, we ask whether the automaton $$\mathcal {A}$$ A specifies some graph satisfying a certain property $$\varPhi $$ Φ . 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 $$F^\infty $$ F ∞ and therefore the family of finite subgraphs of $$F^\infty $$ F ∞ 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.

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“…Finally, between any two occurrences of the factor aaa in w, there must be a factor baab 11. Corollary 4 corrects a misprint in[8, Cor. 3].…”

mentioning

confidence: 82%

Section: Marked Graphsmentioning

confidence: 99%

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Properties of graphs specified by a regular language

2022

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“…The same can be done for string-problems where a given string is to be transformed (by using certain operations) into a target string [4,34]. The int Reg -problem concerning graph problems and graph properties was recently considered in [6,38], where several general decidability results were obtained.…”

Section: Introductionmentioning

confidence: 99%

“…If we consider regular sets of instances, this task can be formalized as checking whether a given regular language of P-instances (represented by a deterministic finite automaton) and the fixed language of positive P-instances have a non-empty intersection. This was the original viewpoint of the line of research introduced by Güler et al [6,10,35,36,38], where this problem is called the int Reg -problem of P (or int Reg (P) for short). 2 The int Reg -problem has independently been studied under the name regular realizability problem R R(L), where the filter language L plays the role of problem P as defined above, i. e., R R(L) = int Reg (L) (see [1, 23-25, 27, 30-32]).…”

Section: Introductionmentioning

confidence: 99%

On the decidability of finding a positive ILP-instance in a regular set of ILP-instances

Wolf

2022

Acta Informatica

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The regular intersection emptiness problem for a decision problem P ($${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg (P)) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since $${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg (P) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the $${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg -problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the $${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg -problem for the well-known NP-complete problem Integer Linear Programming (ILP). It is shown that any DFA that describes a set of ILP-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of $${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg (ILP).

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