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Discussiones Mathematicae Graph Theory 20(2) (2000) 267-269
DOI: https://doi.org/10.7151/dmgt.1125

A NOTE ON PERIODICITY OF THE 2-DISTANCE OPERATOR

Bohdan Zelinka

Department of Applied Mathematics
Technical University of Liberec
Liberec, Czech Republic

To the memory of Ivan Havel

Abstract

The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class C_f of all finite undirected graphs. If G is a graph from C_f, then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.

Keywords: 2-distance operator, complement of a graph.

2000 Mathematics Subject Classification: 05C12.

References

[1]	F. Harary, C. Hoede and D. Kadlacek, Graph-valued functions related to step graphs, J. Comb. Ing. Syst. Sci. 7 (1982) 231-246.
[2]	E. Prisner, Graph Dynamics (Longman House, Burnt Mill, Harlow, 1995).

Received 18 February 2000
Revised 5 July 2000