Dear Colleagues,

We continue our COLLOQUIUM.

Teresa W. Haynes,

East Tennessee State University,

Christine Mynhardt,

University of South Africa,

and Lucas C. van der Merwe,

University of Tennessee at Chattanooga

Tuesday, February 11 , Metro 161, 2:00pm

Total Domination Edge Critical Graphs with Minimum Diameter

Abstract. A set $S$ of vertices of a graph $G$ is a total dominating set
if every vertex of $V(G)$ is adjacent to some vertex in $S$. The total
domination number of $G$, denoted by $\gamma_t(G)$, is the minimum
cardinality of a total dominating set of $G$. A graph $G$ is said to be
total domination edge critical, or simply $\gamma_t$-critical, if
$\gamma_t(G+e)<\gamma_t(G)$ for each edge $e\in E(G)$. For
$3_t$-critical graphs $G$, that is, $\gamma_t$-critical graphs with
$\gamma_t(G)=3$, the diameter of $G$ is either 2 or 3. We study the
$3_t$-critical graphs $G$ with $diam G=2$.