Abstract
We prove that the P 4-transformation is one-to-one on the set of graphs with minimum degree at least 3, and if graphs G and G ' have minimum degree at least 3 then any isomorphism from the P 4-graph P 4(G) to the P 4-graph P 4(G ') is induced by a vertex-isomorphism from G to G ' unless G and G ' both belong to a special family of graphs.
Similar content being viewed by others
References
Aldred, R.E.L., Ellingham, M.N., Hemminger, R.L., Jipsen, P.: P 3-isomorphisms for graphs. J. Graph Theory 26, 35–51 (1997)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications, MacMillan/Elesevier. London/New York, 1976
Broersma, H.J., Hoede, C.: Path graphs. J. Graph Theory 13, 427–444 (1989)
Grünbaum, B.: Incidence patterns of graphs and complexes. Lecture Notes in Mathematics (eds. G. Chartrand, S.F. Kapoor) 110, 115–128 (1969)
Hemminger, R.L., Beineke, L.W.: Line graphs and line digraphs. In: Beineke, L.W., Wilson, R.J. eds. Selected Topics in Graph Theory, Academic Press, London, New York, San Francisco, 1978
Jung, H.A.: Zu einem isomorphiesatz von H. Whitney für graphen. Math. Ann. 164, 270–271 (1966)
Li, H., Lin, Y.: On the charaterization of path graphs. J. Graph Theory 17, 463–466 (1993)
Li, X.: Isomorphisms of P 3-graphs. J. Graph Theory 21, 81–85 (1996)
Li, X.: On the determination problem for P 3-transformation of graphs. Ars Combinatoria 49, 296–302 (1998)
Li, X., Zhao, B.: Isomorphisms of P 4-graphs. Australasian J. Combinatorics 15, 135–143 (1997)
Li, X., Zhao, B.: Isomorphisms of P k -graphs for k ≥ 4. Discrete Math. 289, 71–80 (2004)
Prisner, E.: The dynamics of the line and path graph operators. Graphs and Combinatorics 9, 335–352 (1993)
Prisner, E.: A common generalization of line graphs and clique graphs. J. Graph Theory 18, 301–313 (1994)
Prisner, E.: Line graphs and generalizations — a survey. In: Surveys in Graph Theory (G. Chartrand, M. Jacobson eds.) Congressus Numerantium 116, 193–230 (1996)
Prisner, E.: Bicliques in graphs. II. Recognizing k-path graphs and underlying graphs of line digraphs. Graph-Theoretic Concepts in Computer Science (Berlin, 1997), 273–287, Lecture Notes in Comput. Sci., 1335, Springer, Berlin, 1997
Prisner, E.: Bicliques in graphs I: Bounds on their number. Combinatorica 20(1), 109–117 (2000)
Prisner, E.: Recognizing k-path graphs. Proceedings of the 5th Twente Workshop on Graphs and Combinatorial Optimization (Enschede, 1997), Discrete Appl. Math. 99, 169–181 (2000)
Van Bang Le, Prisner, E.: Iterated k-line graphs. Graphs and Combinatorics 10, 193–203 (1994)
Whitney, H.: Congruent graphs and connectivity of graphs. Amer. J. Math. 54, 150–168 (1932)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC, PCSIRT and the “973” program.
Rights and permissions
About this article
Cite this article
Li, X., Liu, Y. On the Determination Problem for P 4-Transformation of Graphs. Graphs and Combinatorics 24, 551–562 (2008). https://doi.org/10.1007/s00373-008-0808-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-008-0808-6