@article{, AUTHOR = {Favaron, Odile and Karami, Hossein and Khoeilar, Rana and Sheikholeslami, Seyed Mahmoud}, TITLE = {Matchings and total domination subdivision number in graphs with few induced 4-cycles}, JOURNAL = {Discuss. Math. Graph Theory}, FJOURNAL = {Discussiones Mathematicae Graph Theory}, VOLUME = {30}, YEAR = {2010}, NUMBER = {4}, PAGES = {611-618}, ISSN = {1234-3099}, ABSTRACT = {A set $S$ of vertices of a graph $G=(V,E)$ without isolated vertex is a {\em total dominating set} if every vertex of $V(G)$ is adjacent to some vertex in $S$. The {\em total domination number} $\gamma_t(G)$ is the minimum cardinality of a total dominating set of $G$. The {\em total domination subdivision number} ${\rm sd}_{\gamma_t}(G)$ 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 $n\ge 3$, ${\rm sd}_{\gamma_t}(G) \le\gamma_t(G)+1.$ In this paper we use matchings to prove this conjecture for graphs with at most three induced 4-cycles through each vertex. },}