Abstract
In [1] the metric prolongation property (MPP) is introduced for discrete metric spaces, and connections are pointed out of the MPP with metric properties of the locally-isometric embeddings. In the present paper, we continue the study of the MPP for finite graphs [1 – 5]. Some operations invariant with respect to the MPP are considered whose application simplifies the study of the MPP for certain classes of graphs. A theorem is proved on the isometric embedding preserving the MPP of an arbitrary graph to a graph of given connectivity, and it is proved that the problems of describing the classes of connected, those of connectivity n, and of n-connected graphs with he MPP are equivalent. For graphs of small connectivity, in [3,5], and also in the ]present paper, there is obtained a complete characterization of the natural classes of such graphs with the MPP. Conditions are given of the invariance of the MPP for the join operation, and the complete n-partite graphs with the MPP are described. An isometric embedding of an arbitrary graph of diameter 2 to an appropriate graph with the MPP is constructed. Cactii with the MPP are described, which extends the description of trees and unicyclic graphs with the MPP obtained by the author in [3,4].
This research was supported by the Russian Foundation for basic Research (Grant 93-01-01527)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. A. Evdokimov (1988) Metric properties of embeddings, and the codes preserving distances(in Russian), in: Modeli i Melody Optimizatsii. Vol. 10, Trudy Inst. Mat., Nauka, Novosibirsk, pp. 116–132.
A. A. Evdokimov (1994) Locally isometric embeddings of graphs and the metric prolongation property, Sibirsk. Zh. Issled. Oper. 1, No. 1, 5–12.
T. I. Fedoryaeva (1988) Characterization of a class of graphs with the metric prolongation property (in Russian), Metody Diskret. Anal. 47, 89–93.
T. I. Fedoryaeva (1992) Strengthened properties of the metric prolongation (in Russian), Metody Diskret. Anal. 52, 112–118.
T. I. Fedoryaeva (1995) Outerplanar graphs satisfying the metric prolonga tion property (in Russian), Preprint No. 3, Sobolev Institute of Mathematics, Novosibirsk.
V. A. Emelichev, O. I. Mel’nikov, V. I. Sarvanov, and R. I. Tyshkevich (1990) Lectures on Graph Theory (in Russian), Nauka, Moscow.
A. A. Zykov (1987) Fundamentals of Graph Theory (in Russian), Nauka, Moscow.
F. Harary (1969) Graph Theory, Addison-Wesley Publishing Company, Reading, Mass.
F. Harary and E. Palmer (1973) Graphical Enumeration, Academic Press, New York and London.
F. Harary and R. Z. Norman (1953) The dissimilarity characteristic of Husimi trees, Ann. of Math. 58, No. 1, 134–141.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Fedoryaeva, T.I. (1997). Operations and Isometric Embeddings of Graphs Related to the Metric Prolongation Property. In: Operations Research and Discrete Analysis. Mathematics and Its Applications, vol 391. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5678-3_5
Download citation
DOI: https://doi.org/10.1007/978-94-011-5678-3_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6395-1
Online ISBN: 978-94-011-5678-3
eBook Packages: Springer Book Archive