Operations and Isometric Embeddings of Graphs Related to the Metric Prolongation Property

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].