# A linear descriptor for conceptual graphs and a class for polynomial isomorphism test

## Abstract

The isomorphism problem is not known to be NP-complete nor polynomial. Yet it is crucial when maintaining large conceptual graphs databases. Taking advantage of conceptual graphs specificities, whenever, by means of structural functions, a linear order of the conceptual nodes of a conceptual graph *G* can be computed as invariant under automorphism, a descriptor is assigned to *G* in such a way that any other conceptual graph isomorphic to *G* has the same descriptor and conversely. The class of conceptual graphs for which the linear ordering of the conceptual nodes succeeds is compared to other relevant classes, namely those of locally injective, *C*-rigid and irredundant conceptual graphs. Locally injective conceptual graphs are proved to be irredundant, thus linearly ordered by the specialization relation.

## Keywords

Isomorphism problem structural functions descriptors classes of conceptual graphs## Preview

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## References

- [BFH+92]F. Baader, E. Franconi, B. Hollunder, B. Nebel, and H.J. Profitlich. An empirical analysis of optimization techniques for terminological representation systems. In B. Nebel, C. Rich, and W. Swartout, editors,
*Principles of Knowledge Representation and Reasoning — Proceedings of the 3rd International Conference*, Cambridge, MA, 1992.Google Scholar - [CG70]D.G. Corneil and C.C. Gotlieb. An efficient algorithm for graph isomorphism.
*J.A.C.M.*, 17(1):51–64, 1970.Google Scholar - [CK80]D.G. Corneil and D.G. Kirkpatrick. A theorical analysis of various heuristics for the graph isomorphism problem.
*SIAM J. COMPUT.*, 9(2), 1980.Google Scholar - [CM92]M. Chein and M.L. Mugnier. Conceptual graphs: fundamental notions.
*Revue d'Intelligence Artificielle*, 6(4):365–406, 1992.Google Scholar - [Cog76]O. Cogis. A formalism relevant to a classical strategy for the graph isomorphism testing. In
*Proceedings of the 7th S-E Conf. Combinatorics, Graph Theory and Computing*, pages 229–238, Bâton Rouge, 1976.Google Scholar - [EL94]G. Ellis and F. Lehmann. Exploiting the induced order on type-labeled graphs for fast knowledge retrieval. In
*Proceedings of the 2nd Int. Conf. on Conceptual Structures*, Maryland, USA, August 1994. Springer-Verlag. Published as No. 835 of Lecture Notes in Artificial Intelligence.Google Scholar - [Ell91]G. Ellis. Compiled hierarchical retrieval.
*Proceedings of the 6th Annual Workshop on Conceptual Graphs*, pages 187–207, Binghamton, 1991.Google Scholar - [Ell93]G. Ellis. Efficient retrieval from hierarchies of objects using lattice operations. In
*Proceedings of the 1st Int. Conf. on Conceptual Structures*, Montreal, Canada, August 1993. Springer-Verlag. Published as No. 699 of Lecture Notes in Artificial Intelligence.Google Scholar - [Gre82]J.J. Mc Gregor. Backtrack search algorithms and the maximal common subgraph problem.
*Software-Practice and Experience*, 12:23–34, 1982.Google Scholar - [LB94]M. Liquière and O. Brissac. A class of conceptual graphs with polynomial iso-projection. In
*Supplement Proceedings of the 2nd International Conference on Conceptual Structures*, College Park, Maryland, USA, 1994.Google Scholar - [LE91]R. Levinson and G. Ellis. Multi-level hierarchical retrieval. In
*Proceedings of the 6th Annual Workshop on Conceptual Graphs*, pages 67–81, 1991.Google Scholar - [Lev92]R. Levinson. Pattern associativity and the retrieval of semantic networks. In F. Lehmann, editor,
*Semantic Networks in Artificial Intelligence*, pages 573–600, Pergamon Press, Oxford, 1992.Google Scholar - [Lev94]R. Levinson. Uds: A universal data structure. In
*Proceedings of the 2nd Int. Conf. on Conceptual Structures*, Maryland, USA, August 1994. Springer Verlag. Lecture Note in Artificial intelligence #835.Google Scholar - [MC92]M.L. Mugnier and M. Chein. Polynomial algorithms for projection and matching. In Heather D. Pfeiffer, editor,
*Proceedings of the 7th Annual Workshop on Conceptual Graphs*, pages 49–58, New Mexico University, 1992.Google Scholar - [MLL91]S.H. Myaeng and A. Lopez-Lopez. A flexive matching algorithm for matching conceptual graphs. In
*Proceedings of the 6th Annual Workshop on Conceptual Graphs*, pages 135–151, 1991.Google Scholar - [RC77]R.C. Read and D.G. Corneil. The graph isomorphism disease.
*Journal of Graph Theory*, 1:339–363, 1977.Google Scholar - [Sau71]G. Saucier. Un algorithme efficace recherchant l'isomorphisme de deux graphes.
*R.I.R.O.*, 5^{ème}année:39–51, 1971.Google Scholar - [Sir71]F. Sirovich. Isomorfi fra grafi: un algoritmo efficiente per trovare tutti gli isomorfismi.
*Calcolo 8*, pages 301–337, 1971.Google Scholar - [Sow84]J.F. Sowa.
*Conceptual Structures — Information Processing in Mind and Machine*. Addison-Wesley, Reading, Massachusetts, 1984.Google Scholar - [Ung64]S.H. Unger. GIT — a heuristic program for testing pairs of directed line graphs for isomorphism.
*C.A.C.M*, 7(1):26–34, 1964.Google Scholar - [Way94]E. Way. Conceptual graphs — past, present and future. In
*Proceedings of the 2nd Int. Conf. on Conceptual Structures*, Maryland, USA, August 1994. Springer Verlag. Lecture Note in Artificial intelligence #835.Google Scholar - [Woo91]W.A. Woods. Understanding subsomption and taxonomy: A framework for progress. In J.F. Sowa, editor,
*Principles of Semantic Networks, Explorations in the Representation of Knowledge*, pages 45–94, Morgan Kaufmann, San Mateo (USA), 1991.Google Scholar