DMGT

Discussiones Mathematicae Graph Theory 30(4) (2010) 611-618
DOI: https://doi.org/10.7151/dmgt.1517

MATCHINGS AND TOTAL DOMINATION SUBDIVISION NUMBER IN GRAPHS WITH FEW INDUCED 4-CYCLES

Odile Favaron

Univ Paris-Sud, LRI, UMR 8623
Orsay, F-91405, France
CNRS, Orsay, F-91405
e-mail: of@lri.fr

Hossein Karami, Rana Khoeilar and Seyed Mahmoud Sheikholeslami

Department of Mathematics
Azarbaijan University of Tarbiat Moallem
Tabriz, I.R. Iran
e-mail: s.m.sheikholeslami@azaruniv.edu

Abstract

A set S of vertices of a graph G = (V,E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ_t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_{γ_t}(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Favaron, Karami, Khoeilar and Sheikholeslami (Journal of Combinatorial Optimization, to appear) conjectured that: For any connected graph G of order n ≥ 3, sd_{γ_t}(G) ≤ γ_t(G)+1. In this paper we use matchings to prove this conjecture for graphs with at most three induced 4-cycles through each vertex.

Keywords: matching, barrier, total domination number, total domination subdivision number.

2010 Mathematics Subject Classification: 05C69.

>References

[1]	D. Archdeacon, J. Ellis-Monaghan, D. Fisher, D. Froncek, P.C.B. Lam, S. Seager, B. Wei and R. Yuster, Some remarks on domination, J. Graph Theory 46 (2004) 207-210, doi: 10.1002/jgt.20000.
[2]	O. Favaron, H. Karami, R. Khoeilar and S.M. Sheikholeslami, A new upper bound for total domination subdivision numbers, Graphs and Combinatorics 25 (2009) 41-47, doi: 10.1007/s00373-008-0824-6.
[3]	O. Favaron, H. Karami, R. Khoeilar and S.M. Sheikholeslami, On the total domination subdivision number in some classes of graphs, Journal of Combinatorial Optimization, (to appear).
[4]	O. Favaron, H. Karami and S.M. Sheikholeslami, Total domination and total domination subdivision numbers of graphs, Australas. J. Combin. 38 (2007) 229-235.
[5]	O. Favaron, H. Karami and S.M. Sheikholeslami, Bounding the total domination subdivision number of a graph in terms of its order, Journal of Combinatorial Optimization, (to appear).
[6]	T.W. Haynes, M.A. Henning and L.S. Hopkins, Total domination subdivision numbers of graphs, Discuss. Math. Graph Theory 24 (2004) 457-467, doi: 10.7151/dmgt.1244.
[7]	T.W. Haynes, M.A. Henning and L.S. Hopkins, Total domination subdivision numbers of trees, Discrete Math. 286 (2004) 195-202, doi: 10.1016/j.disc.2004.06.004.
[8]	T.W. Haynes, S.T. Hedetniemi and L.C. van der Merwe, Total domination subdivision numbers, J. Combin. Math. Combin. Comput. 44 (2003) 115-128.
[9]	M.A. Henning, L. Kang, E. Shan and A. Yeo, On matching and total domination in graphs, Discrete Math. 308 (2008) 2313-2318, doi: 10.1016/j.disc.2006.10.024.
[10]	H. Karami, A. Khodkar and S.M. Sheikholeslami, An upper bound for total domination subdivision numbers of graphs, Ars Combin. (to appear).
[11]	H. Karami, A. Khodkar, R. Khoeilar and S.M. Sheikholeslami, Trees whose total domination subdivision number is one, Bulletin of the Institute of Combinatorics and its Applications, 53 (2008) 57-67.
[12]	L. Lovász and M.D. Plummer, Matching Theory, Annals of Discrete Math 29 (North Holland, 1886).
[13]	W.T. Tutte, The factorization of linear graphs, J. Lond. Math. Soc. 22 (1947) 107-111, doi: 10.1112/jlms/s1-22.2.107.
[14]	S. Velammal, Studies in Graph Theory: Covering, Independence, Domination and Related Topics, Ph.D. Thesis (Manonmaniam Sundaranar University, Tirunelveli, 1997).
[15]	D.B. West, Introduction to Graph Theory (Prentice-Hall, Inc, 2000).

Received 24 September 2009
Revised 7 January 2010
Accepted 7 January 2010