Afrika Matematika

, Volume 29, Issue 1–2, pp 47–63 | Cite as

PBIB-designs and association schemes arising from minimum bi-connected dominating sets of some special classes of graphs

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Abstract

A dominating set D of a connected graph \(G = (V, E)\) is said to be bi-connected dominating set if the induced subgraphs of both \(\langle D \rangle \) and \(\langle V-D \rangle \) are connected. The bi-connected domination number \(\gamma _{bc}(G)\) is the minimum cardinality of a bi-connected dominating set. A \(\gamma _{bc}\)-set is a minimum bi-connected dominating set of G. In this paper, we obtain the Partially Balanced Incomplete Block (PBIB)-designs with m = 1, 2, 3, 4 and \(\lfloor \frac{p}{2}\rfloor \) association schemes arising from \(\gamma _{bc}\)-sets of some special classes of graphs.

Keywords

Association schemes Partially balanced incomplete block designs Bi-connected dominating sets Strongly regular graph Circulant graph Hypercube 

Mathematics Subject Classification

Primary 05C51 05E30 05C69 

Notes

Acknowledgements

We are immensely grateful to Prof. N. D. Soner for sharing his pearls of wisdom with us, that has greatly improved the manuscript. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

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Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsBangalore UniversityBangaloreIndia

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