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

• Autorzy: Amitava Bhattacharya , Gurusamy Rengasamy Vijayakumar
• Języki publikacji: EN

Abstrakty

EN

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

Słowa kluczowe

EN

domination number; subdivision number; matching

Kategorie tematyczne

• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2002
• Tom: 22
• Numer: 2
• Strony: 335-347

Daty

wydano

2002

Twórcy

autor

Amitava Bhattacharya

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

autor

Gurusamy Rengasamy Vijayakumar

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

Bibliografia

• [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).

Identyfikatory

DOI

10.7151/dmgt.1179