New compact representations of infinite graphs are investigated. Finite automata are used to represent labelled hyper-graphs which can be also multi-graphs. Our approach consists of a general framework where vertices are represented by a regular prefix-free language and edges are represented by a regular language and a function over tuples. We consider three different functions over tuples: given a tuple the first function returns its first difference, the second one returns its suffix and the last one returns its infixes. The first-difference function is substantially a direct generalization to infinite multi-hyper-graphs of the representation introduced by Ehrenfeucht et al. for finite graphs. This representation, though very interesting for finite graphs, turns out to be quite unsatisfactory for infinite graphs. The other two functions we consider while preserving some interesting features of their representation also achieves a high expressive power. As a matter of fact, our formalism either with the suffix or infix function results to be more powerful than the equational graphs introduced by Courcelle and the simple graphs defined by Caucal. The monadic second order theories of these two classes of graphs are undecidable, but still many interesting graph properties are decidable. The use of a regular prefix-free language to represent the vertices allows (fixed the language of the edges) to express a graph by a labelled tree, moreover, the use of finite automata to represent the edges allows the verification of graph properties.

@article{ITA_2001__35_4_311_0, author = {Torre, Salvatore La and Napoli, Margherita}, title = {Automata-based representations for infinite graphs}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, pages = {311--330}, publisher = {EDP-Sciences}, volume = {35}, number = {4}, year = {2001}, zbl = {0997.05067}, mrnumber = {1880802}, language = {en}, url = {http://www.numdam.org/item/ITA_2001__35_4_311_0/} }

TY - JOUR AU - Torre, Salvatore La AU - Napoli, Margherita TI - Automata-based representations for infinite graphs JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2001 DA - 2001/// SP - 311 EP - 330 VL - 35 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/item/ITA_2001__35_4_311_0/ UR - https://zbmath.org/?q=an%3A0997.05067 UR - https://www.ams.org/mathscinet-getitem?mr=1880802 LA - en ID - ITA_2001__35_4_311_0 ER -

Torre, Salvatore La; Napoli, Margherita. Automata-based representations for infinite graphs. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 4, pp. 311-330. http://www.numdam.org/item/ITA_2001__35_4_311_0/

[1] Easy problems for tree-decomposable graphs. J. Algorithms 12 (1991) 308-340. | MR | Zbl

, and ,[2] When Can an Equational Simple Graph Be Generated by Hyperedge Replacement?, in Proc. of the 23th International Symposium on Mathematical Foundations of Computer Science (MFCS'98), edited by L. Brim, J. Gruska and J. Zlatuska. Brno, Czech Republic, August 24-28, 1998, Lecture Notes in Comput. Sci. 1450 (1998) 543-552. | Zbl

,[3] Graph Expressions and Graph Rewritings. Math. Systems Theory 20 (1987) 83-127. | MR | Zbl

and ,[4] The pathwidth and treewidth of cographs. SIAM J. Discrete Math. 6 (1993) 181-188. | MR | Zbl

and ,[5] On infinite transition graphs having a decidable monadic theory, in Proc. of 23rd International Colloquium on Automata, Languages and Programming (ICALP'96), edited by F.M. auf der Heide and B. Monien. Paderborn, Germany, July 8-12, 1996, Lecture Notes in Comput. Sci. 1099 (1996) 194-205. | MR | Zbl

,[6] Complement reducible graphs. Discrete Appl. Math. 3 (1981) 163-174. | MR | Zbl

, and ,[7] The monadic second-order logic of graphs. II. Infinite graphs of bounded width. Mathematical Systems Theory 21 (1989) 187-221. | MR | Zbl

,[8] The monadic second-order logic of graphs. III. Tree-width, forbidden minors and complexity issues. RAIRO: Theoret. Informatics Appl. 26 (1992) 257-286. | EuDML | Numdam | MR | Zbl

,[9] Finite Languages for the Representation of Finite Graphs. J. Comput. System Sci. 52 (1996) 170-184. | MR | Zbl

, and ,[10] Characterization and Complexity of Uniformly Non-primitive Labeled 2-Structures. Theoretical Comput. Sci. 154 (1996) 247-282. | MR | Zbl

, , and ,[11] Multi-Tape and Infinite-State Automata - A Survey. Comm. ACM 8 (1965) 799-805. | Zbl

,[12] Introduction to Automata Theory, Formal Languages and Computation. Addison-Wesley Publishing Company, Addison-Wesley Series in Comput. Sci.(1979) | MR | Zbl

and ,[13] Representing Hyper-Graphs by Regular Languages. in Proc. of the 23th International Symposium on Mathematical Foundations of Computer Science (MFCS'98), edited by L. Brim, J. Gruska and J. Zlatuska, Brno, Czech Republic, August 24-28, 1998, Lecture Notes in Comput. Sci. 1450 (1998) 571-579. | Zbl

and ,[14] On Rational Graphs, in Proc. of the 3rd International Conference on Foundations of Software Science and Computation Structures (FOSSACS 2000). Berlin, Germany, March 25 - April 2, 2000, Lecture Notes in Comput. Sci. 1784 (2000) 252-266. | MR | Zbl

,[15] Graph Minors. II Algorithmic aspects of tree-width. J. Algorithms 7 (1986) 309-322. | MR | Zbl

and ,[16] The recognition of series parallel digraphs. SIAM J. Comput. 11 (1982) 298-313. | MR | Zbl

, and ,