Matchings and total domination subdivision number in graphs with few induced 4-cycles

Odile Favaron ; Hossein Karami ; Rana Khoeilar ; Seyed Mahmoud Sheikholeslami

Discussiones Mathematicae Graph Theory, Tome 30 (2010), p. 611-618 / Harvested from The Polish Digital Mathematics Library

Résumé

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 γₜ(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number $sd_{γₜ(G)}$ 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, $sd_{γₜ(G)} ≤ γₜ(G)+1$. In this paper we use matchings to prove this conjecture for graphs with at most three induced 4-cycles through each vertex.

Détail

Publié le : 2010-01-01

Zbl 1217.05178

EUDML-ID : urn:eudml:doc:270987

Bibliographie

[000] [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. | Zbl 1041.05057

[001] [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. | Zbl 1211.05109

[002] [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). | Zbl 1198.05121

[003] [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. | Zbl 1133.05068

[004] [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). | Zbl 1213.90212

[005] [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. | Zbl 1065.05070

[006] [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. | Zbl 1054.05076

[007] [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. | Zbl 1020.05048

[008] [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. | Zbl 1145.05042

[009] [10]. H. Karami, A. Khodkar and S.M. Sheikholeslami, An upper bound for total domination subdivision numbers of graphs, Ars Combin. (to appear). | Zbl 1265.05462

[010] [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. | Zbl 1168.05050

[011] [12]. L. Lovász and M.D. Plummer, Matching Theory, Annals of Discrete Math 29 (North Holland, 1886).

[012] [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. | Zbl 0029.23301

[013] [14]. S. Velammal, Studies in Graph Theory: Covering, Independence, Domination and Related Topics, Ph.D. Thesis (Manonmaniam Sundaranar University, Tirunelveli, 1997).

[014] [15]. D.B. West, Introduction to Graph Theory (Prentice-Hall, Inc, 2000).