# Effect of edge-subdivision on vertex-domination in a graph

Amitava Bhattacharya; Gurusamy Rengasamy Vijayakumar

Discussiones Mathematicae Graph Theory (2002)

- Volume: 22, Issue: 2, page 335-347
- ISSN: 2083-5892

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topAmitava Bhattacharya, and Gurusamy Rengasamy Vijayakumar. "Effect of edge-subdivision on vertex-domination in a graph." Discussiones Mathematicae Graph Theory 22.2 (2002): 335-347. <http://eudml.org/doc/270539>.

@article{AmitavaBhattacharya2002,

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 ln n+5},

author = {Amitava Bhattacharya, Gurusamy Rengasamy Vijayakumar},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {domination number; subdivision number; matching},

language = {eng},

number = {2},

pages = {335-347},

title = {Effect of edge-subdivision on vertex-domination in a graph},

url = {http://eudml.org/doc/270539},

volume = {22},

year = {2002},

}

TY - JOUR

AU - Amitava Bhattacharya

AU - Gurusamy Rengasamy Vijayakumar

TI - Effect of edge-subdivision on vertex-domination in a graph

JO - Discussiones Mathematicae Graph Theory

PY - 2002

VL - 22

IS - 2

SP - 335

EP - 347

AB - 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 ln n+5

LA - eng

KW - domination number; subdivision number; matching

UR - http://eudml.org/doc/270539

ER -

## References

top- [1] N. Alon and J. H. Spencer, The Probabilistic Method, Second Edition, John Wiley and Sons Inc. (Tel Aviv and New York, 2000). Zbl0996.05001
- [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. Zbl0984.05066
- [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. Zbl1006.05042
- [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Dekker, New York, 1998). Zbl0890.05002
- [6] S. Velammal, Studies in Graph Theory: Covering, Independence, Domination and Related Topics, Ph.D. Thesis (Manonmaniam Sundaranar University, Tirunelveli, 1997).

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