# On the total domination subdivision numbers in graphs

Open Mathematics (2010)

- Volume: 8, Issue: 3, page 468-473
- ISSN: 2391-5455

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topSeyed Sheikholeslami. "On the total domination subdivision numbers in graphs." Open Mathematics 8.3 (2010): 468-473. <http://eudml.org/doc/269209>.

@article{SeyedSheikholeslami2010,

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 sdγt (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. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727–733) proved that for any connected graph G of order n ≥ 3, sdγ t(G) ≤ 2γ t(G) − 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sdγ t(G) ≤ 2α′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sdγ t(G)=2γt(G)−1.},

author = {Seyed Sheikholeslami},

journal = {Open Mathematics},

keywords = {Total domination number; Total domination subdivision number; total domination number; total domination subdivision number},

language = {eng},

number = {3},

pages = {468-473},

title = {On the total domination subdivision numbers in graphs},

url = {http://eudml.org/doc/269209},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Seyed Sheikholeslami

TI - On the total domination subdivision numbers in graphs

JO - Open Mathematics

PY - 2010

VL - 8

IS - 3

SP - 468

EP - 473

AB - 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 sdγt (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. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727–733) proved that for any connected graph G of order n ≥ 3, sdγ t(G) ≤ 2γ t(G) − 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sdγ t(G) ≤ 2α′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sdγ t(G)=2γt(G)−1.

LA - eng

KW - Total domination number; Total domination subdivision number; total domination number; total domination subdivision number

UR - http://eudml.org/doc/269209

ER -

## References

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- [9] Karami H., Khodkar A., Khoeilar R., Sheikholeslami S.M., Trees whose total domination subdivision number is one, Bull. Inst. Combin. Appl., 2008, 53, 57–67 Zbl1168.05050
- [10] Karami H., Khodkar A., Khoeilar R., Sheikholeslami S.M., An upper bound for the total domination subdivision number of a graph, Graphs Combin., 2009, 25, 727–733 http://dx.doi.org/10.1007/s00373-010-0877-1 Zbl1205.05169
- [11] Karami H., Khodkar A., Sheikholeslami S.M., An upper bound for total domination subdivision numbers of graphs, Ars Combin., to appear Zbl1265.05462
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