Abstract
We study planar graphs embedded in the plane that have chemical applications: the degrees of all vertices are 3 or 2, all internal faces but one or two arer-gons, and each internal face is a simply connected domain. For wide classes of such graphs, we solve the existence problem for embeddings of the graph metric on the vertices in multidimensional cubes or cubical lattices preserving or doubling all the distances. Incidentally we present a complete classification of some interesting families of such graphs.
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M. Deza and M. Shtogrin, “Polycycles,” in:Voronoi Conference on Analytic Number Theory and Space Tilings (Kiev, September 7–14, 1998), Abstracts, Kiev (1998), pp. 19–23.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, E. Brendsdal, and Fuji Zhang, Xiofeng Guo, R. Tosic, “Theory of polypentagons,”J. Chem. Inf. Comput. Sci.,33, 466–474 (1993).
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, “General formulations for some polycyclic hydrocarbons: di-q-polyhexes,”Chemical Physics Letters,240, 601–604 (1995).
J. R. Dias, “Indancenoid isomers of semibuckmisterfullerene (Buckybowl) and their topological characteristics,”J. Chem. Inf. Comput. Sci.,35, 148–151 (1995).
J. Brunvoll, S. J. Cyvin, and B. N. Cyvin, “Azulenoids,”MATCH,34, 91–108 (1996).
V. Chepoi, M. Deza, and V. Grishukhin, “Clin d’oeil onL 1-embeddable planar graphs,”Discrete Appl. Math.,80, 3–19 (1997).
M. Deza and S. Shpectorov, “Recognition ofl 1-graphs with complexityO(nm) or football in a hypercube,”Europ. J. Combinatorics,17, 279–289 (1996).
M. Deza and M. I. Shtogrin, “Isometric embeddings of semiregular polyhedra and their dual in hypercubes and cubical lattices,”Uspekhi Mat. Nauk [Russian Math. Surveys],51, No. 6, 199–200 (1996).
S. Klavzar, I. Gutman, and B. Mohar, “Labelling of bensenoid systems which reflects the vertex-distance relations,”J. Chem. Inf. Comput. Sci.,35, 590–593 (1995).
M. E. Tylkin, “On the Hamming geometry of unit cubes,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],134 No. 5, 1037–1040 (1960).
K. F. Prisakar’, P. S. Soltan, and V. D. Chepoi, “On embeddability of planar graphs in hypercubes,”Izv. Mold. Akad. Nauk Ser. Mat. [Moldavian Acad. Sci. Izv. Math.], No. 1, 43–50 (1990).
M. Deza and J. Tuma, “A note onl 1-rigid planar graphs,”Europ. J. Combinatorics,17, No. 2, 3, 157–160 (1996).
M. Deza and M. I. Shtogrin, “Primitive polycycles and helicenes”,Uspekhi Mat. Nauk [Russian Math. Surveys],54, No. 6, 159–160 (1999).
M. I. Shtogrin, “Primitive polycycles: a test,”Uspekhi Mat. Nauk [Russian Math. Surveys],54, No. 6, 177–178 (1999).
A. D. Aleksandrov,Convex Polyhedra [in Russian], Gostekhizdat, Moscow-Leningrad (1950).
J. R. Dias,Handbook of Polycyclic Hydrocarbons. Part B: Polycyclic Isomers and Heteroatom Analogs of Bensenoid Hydrocarbons, Elsevier (1988).
A. Dress and G. Brinkmann, “Phantasmagorical fulleroids,”MATCH,33, 87–100 (1996).
H. Hosoya, Y. Okuma, Y. Tsukano, and K. Nakada, “Multilayered cyclic fence graphs: novel cubic graphs related to the graphite network,”J. Chem. Inf. Comput. Sci.,35, 351–356 (1995).
A. Császárs, “A polyhedron without diagonals,”Acta Sci. Math.,13, 140–142 (1949).
A. Nakamoto, “Irreducible quadrangulations of the torus,”J. Combin. Theory. Ser. B,67, 183–201 (1996).
A. Nakamoto, “Irreducible quadrangulations of the Klein bottle,”Yokohama Math. J.,43, 125–139 (1995)
A. T. Balaban, “Chemical graphs: Looking back and glimpsing ahead,”J. Chem. Inf. Comput. Sci.,35, No. 3, 345 (1995).
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Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 339–352, September, 2000.
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Deza, M., Shtogrin, M.I. Embeddings of chemical graphs in hypercubes. Math Notes 68, 295–305 (2000). https://doi.org/10.1007/BF02674552
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DOI: https://doi.org/10.1007/BF02674552