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. },