Abstract
Given a directed graph $D$, a set $S \subseteq V(D)$ is a total dominating
set of $D$ if each vertex in $D$ has an in-neighbor in $S$. The total
domination number of $D$, denoted $\gamma_t(D)$, is the minimum cardinality
among all total dominating sets of $D$. Given an undirected graph $G$, we study
the maximum and minimum total domination numbers among all orientations of $G$.
That is, we study the upper (or lower) orientable domination number of $G$,
$\rm{DOM}_t(G)$ (or $\rm{dom}_t(G)$), which is the largest (or smallest) total
domination number over all orientations of $G$. We characterize those graphs
with $\rm{DOM}_t(G) =\rm{dom}_t(G)$ when the girth is at least $7$ as well as
those graphs with $\rm{dom}_t(G) = |V(G)|-1$. We also consider how these
parameters are effected by removing a vertex from $G$, give exact values of
$\rm{DOM}_t(K_{m,n})$ and $\rm{dom}_t(K_{m,n})$ and bound these parameters when
$G$ is a grid graph.