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

• Volume: 30, Issue: 4, page 611-618
• ISSN: 2083-5892

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## Abstract

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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 $s{d}_{\gamma ₜ\left(G\right)}$ 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, $s{d}_{\gamma ₜ\left(G\right)}\le \gamma ₜ\left(G\right)+1$. In this paper we use matchings to prove this conjecture for graphs with at most three induced 4-cycles through each vertex.

## How to cite

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Odile Favaron, et al. "Matchings and total domination subdivision number in graphs with few induced 4-cycles." Discussiones Mathematicae Graph Theory 30.4 (2010): 611-618. <http://eudml.org/doc/270987>.

@article{OdileFavaron2010,
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 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.},
author = {Odile Favaron, Hossein Karami, Rana Khoeilar, Seyed Mahmoud Sheikholeslami},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {matching; barrier; total domination number; total domination subdivision number},
language = {eng},
number = {4},
pages = {611-618},
title = {Matchings and total domination subdivision number in graphs with few induced 4-cycles},
url = {http://eudml.org/doc/270987},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Odile Favaron
AU - Hossein Karami
AU - Rana Khoeilar
AU - Seyed Mahmoud Sheikholeslami
TI - Matchings and total domination subdivision number in graphs with few induced 4-cycles
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 4
SP - 611
EP - 618
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 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.
LA - eng
KW - matching; barrier; total domination number; total domination subdivision number
UR - http://eudml.org/doc/270987
ER -

## References

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11. [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. Zbl1168.05050
12. [12]. L. Lovász and M.D. Plummer, Matching Theory, Annals of Discrete Math 29 (North Holland, 1886).
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