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Shuffles and concatenations in the construction of graphs†

Published online by Cambridge University Press:  30 October 2012

KOSTA DOŠEN  and
ZORAN PETRIĆ
KOSTA DOŠEN
Affiliation:
Mathematical Institute, SANU, Knez Mihailova 36, P.O. Box 367, 11001 Belgrade, Serbia Email: kosta@mi.sanu.ac.rs, zpetric@mi.sanu.ac.rs
ZORAN PETRIĆ
Affiliation:
Mathematical Institute, SANU, Knez Mihailova 36, P.O. Box 367, 11001 Belgrade, Serbia Email: kosta@mi.sanu.ac.rs, zpetric@mi.sanu.ac.rs
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Abstract

This paper reports on an investigation into the role of shuffling and concatenation in the theory of graph drawing. A simple syntactic description of these and related operations is proved to be complete in the context of finite partial orders, and as general as possible. An explanation based on this result is given for a previously investigated collapse of the permutohedron into the associahedron, and for collapses into other less familiar polyhedra, including the cyclohedron. Such polyhedra have been considered recently in connection with the notion of tubing, which is closely related to tree-like finite partial orders, which are defined simply here and investigated in detail. Like the associahedron, some of these other polyhedra are involved in categorial coherence questions, which will be treated elsewhere.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Issue 6: CAMCAD '09 Commutativity of Algebraic Diagrams , December 2012 , pp. 904 - 930
Copyright
Copyright © Cambridge University Press 2012

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Footnotes

†

Work on this paper was supported by the Ministry of Science of Serbia (Grants 144013 and 144029).

References

Aceto, L. and Fokkink, W. (2006) The quest for equational axiomatizations of parallel composition: Status and open problems. Electronic Notes in Theoretical Computer Science 162 43–48.CrossRefGoogle Scholar
Armstrong, S., Carr, M., Devadoss, S. L., Eugler, E., Leininger, A. and Manapat, M. (2009) Particle configurations and Coxeter operads. Journal of Homotopy and Related Structures 4 83–109.Google Scholar
Bloom, J. M. (2011) A link surgery spectral sequence in monopole Floer homology. Advances in Mathematics 226 3216–3281.CrossRefGoogle Scholar
Carr, M. and Devadoss, S. L. (2006) Coxeter complexes and graph-associahedra. Topology and its Applications 153 2155–2168.CrossRefGoogle Scholar
Devadoss, S. L. (2009) A realization of graph-associahedra. Discrete Mathematics 309 271–276.CrossRefGoogle Scholar
Devadoss, S. L. and Forcey, S. (2008) Marked tubes and the graph multiplihedron. Algebraic and Geometric Topology 8 2084–2108.CrossRefGoogle Scholar
Došen, K. and Petrić, Z. (2006) Associativity as commutativity. Journal of Symbolic Logic 71 217–226 (also available at arXiv).CrossRefGoogle Scholar
Došen, K. and Petrić, Z. (2011) Hypergraph polytopes. Topology and its Applications 158 1405–1444 (also available at arXiv).CrossRefGoogle Scholar
Došen, K. and Petrić, Z. (2010) Weak Cat-operads (preprint available at arXiv).Google Scholar
Forcey, S. and Springfield, D. (2010) Geometric combinatorial algebras: Cyclohedron and simplex. Journal of Algebraic Combinatorics 32 597–627.CrossRefGoogle Scholar
Gischer, J. L. (1988) The equational theory of pomsets. Theoretical Computer Science 62 299–224.Google Scholar
Grabowski, J. (1981) On partial languages. Fundamenta Informaticae 4 427–498†.CrossRefGoogle Scholar
Harary, F. (1969) Graph Theory, Addison-Wesley.CrossRefGoogle Scholar
Jech, T. J. (1973) The Axiom of Choice, North-Holland.Google Scholar
Joni, S. and Rota, G. (1979) Coalgebras and bialgebras in combinatorics. Studies in Applied Mathematics 61 93–139.CrossRefGoogle Scholar
MacLane, S. Lane, S. (1998) Categories for the Working Mathematician (expanded second edition), Springer-Verlag.Google Scholar
Postnikov, A. (2009) Permutohedra, associahedra, and beyond. International Mathematics Research Notices 2009 1026–1106.CrossRefGoogle Scholar
Postnikov, A., Reiner, V. and Williams, L. (2008) Faces of generalized permutohedra. Documenta Mathematica 13 207–273.CrossRefGoogle Scholar
Riguet, J. (1948) Relations binaires, fermetures, correspondances de Galois. Bulletin de la Société Mathématique de France 76 114–155.CrossRefGoogle Scholar
Schmidt, G. and Ströhlein, T. (1993) Relations and Graphs: Discrete mathematics for Computer Scientists, Springer-Verlag.CrossRefGoogle Scholar
Stasheff, J. (1997) From operads to physically inspired theories (Appendix B co-authored with S. Shnider). In: Loday, J.-L., Stasheff, J. D. and Voronov, A. A. (eds.) Operads: Proceedings of Renaissance Conferences. Contemporary Mathematics 202, American Mathematical Society 53–81.Google Scholar
Tonks, A. (1997) Relating the associahedron and the permutohedron. In: Loday, J.-L., Stasheff, J. D. and Voronov, A. A. (eds.) Operads: Proceedings of Renaissance Conferences. Contemporary Mathematics 202, American Mathematical Society 33–36.Google Scholar
Tschantz, S. T. (1994) Languages under concatenation and shuffling. Mathematical Structures in Computer Science 4 505–511.CrossRefGoogle Scholar
Valdes, J., Tarjan, R. E. and Lawler, E. L. (1982) The recognition of series parallel digraphs. SIAM Journal on Computing 11 298–313.CrossRefGoogle Scholar