Discussiones Mathematicae Graph Theory 28(3) (2008) 393-418
doi: 10.7151/dmgt.1415

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Ondrej Vacek

Department of Mathematics and Descriptive Geometry
Faculty of Wood Sciences and Technology
Technical University Zvolen
T.G. Masaryka 24, 960 53 Zvolen, Slovak Republic
e-mail: o.vacek@vsld.tuzvo.sk


The concepts of critical and cocritical radius edge-invariant graphs are introduced. We prove that every graph can be embedded as an induced subgraph of a critical or cocritical radius-edge-invariant graph. We show that every cocritical radius-edge-invariant graph of radius r ≥ 15 must have at least 3r+2 vertices.

Keywords: extremal graphs, radius of graph.

2000 Mathematics Subject Classification: 05C12, 05C35.


[1] V. Bálint and O. Vacek, Radius-invariant graphs, Math. Bohem. 129 (2004) 361-377.
[2] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, 1990).
[3] R.D. Dutton, S.R. Medidi and R.C. Brigham, Changing and unchanging of the radius of graph, Linear Algebra Appl. 217 (1995) 67-82, doi: 10.1016/0024-3795(94)00153-5.
[4] F. Gliviak, On radially extremal graphs and digraphs, a survey, Math. Bohem. 125 (2000) 215-225.
[5] S.M. Lee, Design of diameter e-invariant networks, Congr. Numer. 65 (1988) 89-102.
[6] S.M. Lee and A.Y. Wang, On critical and cocritical diameter edge-invariant graphs, Graph Theory, Combinatorics, and Applications 2 (1991) 753-763.
[7] O. Vacek, Diameter-invariant graphs, Math. Bohem. 130 (2005) 355-370.
[8] V.G. Vizing, On the number of edges in graph with given radius, Dokl. Akad. Nauk 173 (1967) 1245-1246 (in Russian).
[9] H.B. Walikar, F. Buckley and K.M. Itagi, Radius-edge-invariant and diameter-edge-invariant graphs, Discrete Math. 272 (2003) 119-126, doi: 10.1016/S0012-365X(03)00189-4.

Received 11 September 2006
Revised 18 December 2007
Accepted 6 June 2008