DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 22(2) (2002) 335-347
DOI: 10.7151/dmgt.1179

EFFECT OF EDGE-SUBDIVISION ON VERTEX-DOMINATION IN A GRAPH

 Amitava Bhattacharya and Gurusamy Rengasamy Vijayakumar

School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road, Colaba
Mumbai 400 005, India

e-mail: amitava@math.tifr.res.in
e-mail: vijay@math.tifr.res.in

Abstract

Let G be a graph with Δ(G) > 1. It can be shown that the domination number of the graph obtained from G by subdividing every edge exactly once is more than that of G. So, let ξ(G) be the least number of edges such that subdividing each of these edges exactly once results in a graph whose domination number is more than that of G. The parameter ξ(G) is called the subdivision number of G. This notion has been introduced by S. Arumugam and S. Velammal. They have conjectured that for any graph G with Δ(G) > 1, ξ(G) ≤ 3. We show that the conjecture is false and construct for any positive integer n ≥ 3, a graph G of order n with ξ(G) > [1/3]log2 n. The main results of this paper are the following: (i) For any connected graph G with at least three vertices, ξ(G) ≤ γ(G)+1 where γ(G) is the domination number of G. (ii) If G is a connected graph of sufficiently large order n, then ξ(G) ≤ 4√n lnn+5.

 Keywords: domination number, subdivision number, matching.

 2000 Mathematics Subject Classification: 05C69.

References

[1] N. Alon and J. H. Spencer, The Probabilistic Method, Second Edition, John Wiley and Sons Inc. (Tel Aviv and New York, 2000).
[2] R. Diestel, Graph Theory, Second Edition (Springer-Verlag, New York, 2000).
[3] T.W. Haynes, S.M. Hedetniemi and S.T. Hedetniemi, Domination and independence subdivision numbers of graphs, Discuss. Math. Graph Theory 20 (2000) 271-280, doi: 10.7151/dmgt.1126.
[4] T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, D.P. Jacobs, J. Knisely and L.C. van der Merwe, Domination Subdivision Numbers, preprint.
[5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Dekker, New York, 1998).
[6] S. Velammal, Studies in Graph Theory: Covering, Independence, Domination and Related Topics, Ph.D. Thesis (Manonmaniam Sundaranar University, Tirunelveli, 1997).
Received 8 June 2001
Revised 20 October 2001