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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: [email protected]

Hossein Karami, Rana Khoeilar and Seyed Mahmoud Sheikholeslami

Department of Mathematics
Azarbaijan University of Tarbiat Moallem
Tabriz, I.R. Iran
e-mail: [email protected]

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

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