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On Edge Semi-Isomorphisms and Semi-Dualities of Graphs

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Abstract

Let G1 and G2 be undirected graphs, and ℱ1(G 1) and ℱ2(G 2) be families of edge sets of G1 and G2, respectively. An (ℱ1,ℱ2)-semi-isomorphism ofG 1 ontoG 2 is an edge bijection between G1 and G2 that induces an injection from ℱ1(G 1) to ℱ2(G 2). This concept generalizes a well known concept of a circuit isomorphism of graphs due to H. Whitney. If has a “dual nature” with respect to ℱ2(G 2) then the concept of (ℱ1,ℱ2)-semi-isomorphism of graphs turns into a concept of a (ℱ1,ℱ2)-semi-duality of graphs. This gives a natural generalization of the circuit duality of graphs due to H. Whitney. In this paper we investigate (ℱ1,ℱ2)-semi-isomorphisms and (ℱ1,ℱ2)-semi-dualities of graphs for various families ℱ1(G 1) and ℱ2(G 2). In particular, we consider families of circuits and cocircuits of graphs from this point of view, and obtain some strengthenings of Whitney’s 2-isomorphism theorem and Whitney’s planarity criterion for 3-connected graphs.

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References

  1. Barnette, D.: Generating the triangulations of the projective plane, J. Comb. Theory 33,222–230(1982)

    MATH  MathSciNet  Google Scholar 

  2. Berge, Theory of Graphs and its applications, Methuen London, 1962

  3. Bondy, J.A., Murty, U.S.R.: Graph theory and applications, MacMillan Co, London, 1976

    Google Scholar 

  4. Halin, R., Jung, H.A.: Note on isomorphism of grahs, J. London Math. Soc. 42, 254–256 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hemminger, R.L., Jung, H.A., Kelmans, A.K.: 3-skein isomorphism of graphs, Combinatorica 2(4), 373–376 (1982)

    MATH  MathSciNet  Google Scholar 

  6. Hemminger, R.L., Jung, H.A.: On n-skein isomorphism of graphs, J. Comb. Theory 32,103–111(1982)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hemminger, R.L., Jung, H.A.: On 3-skein injection of graphs, Congressus Numerantium 36,347–350(1982)

    MathSciNet  Google Scholar 

  8. Kelmans, A.K.: A concept of a vertex in a matroid, the non-separating circuits of a graph and a new criterion for graph planarity, in: Algebraic methods in graph theory (Szeged, Hungary, (1978) Amsterdam, Budapest (1981) 345–387

  9. Kelmans, A.K.: The concept of a vertex in a matroid and 3-connected graphs, J. Graph Theory 4, 13–19(1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kelmans, A.K.: 3-skeins in a 3-connected graph, in: Vsesouznoe soveschanie po statisticheskomu i diskretnomu analizu, Kazanskiy gos. Universitet, Moskva-Alma-Ata (1981) (in Russian)

  11. Kelmans, A.K.: On edge maps of graphs preserving the subgraphs of a given type, in: Modely i algoritmy issledovaniya operaciy i ih primenenie organizacii raboty v vychiclitenyh systemah, Yaroslavskiy gos. Universitet, Yaroslavl (1984) 19–30

  12. Kelmans, A.K.: On homeomorphic imbedding of graphs with given properties, Doklady AN SSSR 274(6), 2998–3003 (1984)

    MathSciNet  Google Scholar 

  13. Kelmans, A.K.: On 3-connected graphs without essential 3-cuts or triangles, Doklady AN SSSR 288(3), 698–703 (1986)

    MathSciNet  Google Scholar 

  14. Kelmans, A.K.: On 3-skeins in a 3-connected graphs, Studia Scientiarum Mathematicarum Hungarica 22, 265–273 (1987)

    MATH  MathSciNet  Google Scholar 

  15. Kelmans, A.K.: A short proof and a strengthening of the Whitney 2-isomorphism theorem on graphs, Discrete Math. 64, 13–25 (1987)

    MATH  MathSciNet  Google Scholar 

  16. Kelmans, A.K.: Matroids and Whitney’s theorems on 2-isömorphism and planarity of graphs, Uspehi Mat. Nauk 43(5), 199–200 (1988) (in Russian)

    MathSciNet  Google Scholar 

  17. Kelmans, A.K.: On edge semi-isomorphism and semi-dualities of graphs, RUTCOR Research Report 59-90, Rutgers University (1990) 1–15

  18. Sanders, J.H., Sanders, D.: Circuit preserving edge maps, J. Comb. Theory (B) 22, 91–96(1977)

    Article  MATH  MathSciNet  Google Scholar 

  19. Truemper, K.: On Whitney’s 2-isomorphism theorem for graphs, J. Graph Theory 4, 43–49 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  20. Welsh, D.: Matroid theory, Academic Press, London, 1976

    MATH  Google Scholar 

  21. Whitney, H.: Congruent graphs and the connectivity of graphs, Amer. Math. J. 54, 150–168(1932)

    Article  MathSciNet  Google Scholar 

  22. Whitney, H.: Non-separable and planar graphs, Trans. Amer. Math. Soc. 34, 339–362(1932)

    Article  MathSciNet  Google Scholar 

  23. Whitney, H.: 2-isomorphic graphs, Amer. Math. J. 55, 245–254 (1933)

    Article  MathSciNet  Google Scholar 

  24. Whitney, H.: Planar graphs, Fund. Math. 21, 73–84 (1933)

    Google Scholar 

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Authors and Affiliations

  1. RUTCOR, Rutgers University, 08903, New Brunswick, NJ

    A. K. Kelmans

  2. Department of Mathematics, University of Puerto Rico, 00931, San Juan, PR, USA

    A. K. Kelmans

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  1. A. K. Kelmans
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Kelmans, A.K. On Edge Semi-Isomorphisms and Semi-Dualities of Graphs. Graphs and Combinatorics 10, 337–352 (1994). https://doi.org/10.1007/BF02986684

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  • DOI: https://doi.org/10.1007/BF02986684

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