ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


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


 Amitava Bhattacharya and Gurusamy Rengasamy Vijayakumar

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



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.


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Received 8 June 2001
Revised 20 October 2001