# Minor searching, normal forms of graph relabelling: Two applications based on enumerations by graph relabelling

Conference paper

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## Abstract

This paper deals with graph relabelling introduced in [LMS95]. Our first result concerns the open problem of searching a graph as a minor in a graph with a distinguished vertex, by means of graph relabellings. We give and prove a graph rewriting system which answers to this problem. Secondly we define and study normal forms of graph relabellings. We prove that any graph rewriting system can be simulated by a system in k-normal form (with an integer k depending on the original system). Proofs for both results are linked by the enumeration systems they used.

## Key-words

Local computations graph relabelling enumerations paths minor normal form of graph rewritings Download
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## References

- [Ang80]D. Angluin. Local and global properties in networks of processors. In
*12*^{th}*STOC*, pages 82–93, 1980.Google Scholar - [BM96]A. Bottreau and Y. Métivier. Kronecker product and local computation in graphs. In
*CAAP'96*, volume 1059 of*Lect. Notes in Comp. Sci.*, pages 2–16, 1996.Google Scholar - [Bot97]A. Bottreau.
*Réécritures de graphe et calculs distribués*. PhD thesis, Université Bordeaux I, LaBEI, juin 1997.Google Scholar - [CM94]B. Courcelle and Y. Métivier. Coverings and minors: Application to local computations in graphs.
*Europ. J. Combinatorics*, 15:127–138, 1994.CrossRefGoogle Scholar - [Cou90]B. Courcelle. The monadic second order logic of graphs i. recognizable sets of finite graphs.
*Inform, and Comput.*, 85:12–75, 1990.MATHCrossRefMathSciNetGoogle Scholar - [LM92]I. Litovsky and Y. Métivier. Computing trees with graph rewriting systems with priorities.
*Tree Automata and Languages*, pages 115–139, 1992.Google Scholar - [LM93]I. Litovsky and Y. Métivier. Computing with graph rewriting systems with priorities.
*Theoretical Computer Science*, 115:191–224, 1993.CrossRefMathSciNetGoogle Scholar - [LMS95]I. Litovsky, Y. Métivier, and E. Sopena. Different local controls for graph relabelling systems.
*Mathematical Systems Theory*, 28:41–65, 1995.CrossRefMathSciNetGoogle Scholar - [LMZ95]I. Litovsky, Y. Métivier, and W. Zielonka. On the recognition of families of graphs with local computations.
*Information and computation*, 115(1):110–119, 1995.CrossRefGoogle Scholar - [Maz87]A. Mazurkiewicz.
*Petri nets, applications and relationship to other models of concurrency*, volume 255, chapter Trace Theory, pages 279–324. W. Brauer et al., 1987.Google Scholar - [RFH72]P. Rosensthiel, J.R. Fiksel, and A. Holliger. Intelligent graphs: networks of finite automata capable of solving graph problems. In
*Graph Theory and Computing*, pages 219–265. Academic Press, 1972.Google Scholar - [RS95]N. Robertson and P.D. Seymour. Graph minors xiii. the disjoint paths problem.
*Journal of combinatorial theory, Series B*, 63:65–110, 1995.CrossRefMathSciNetGoogle Scholar - [YK96a]M. Yamashita and T. Kameda. Computing on anonymous networks: Part i 3-characterizing the solvable cases.
*IEEE Transactions on parallel and distributed systems*, 7(1):69–89, 1996.CrossRefGoogle Scholar - [YK96b]M. Yamashita and T. Kameda. Computing on anonymous networks: Part ii — decision and membership problems.
*IEEE Transactions on parallel and distributed systems*, 7(1):90–96, 1996.CrossRefGoogle Scholar

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