Discussiones Mathematicae Graph Theory 30(4) (2010) 611-618
doi: 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 γt(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. Favaron, Karami, Khoeilar and Sheikholeslami (Journal of Combinatorial Optimization, to appear) conjectured that: For any connected graph G of order n ≥ 3, sdγt(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.

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

2010 Mathematics Subject Classification: 05C69.

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