  A sequence of vertices in a graph $G$ with no isolated vertices is called a
total dominating sequence if every vertex in the sequence totally dominates at
least one vertex that was not totally dominated by preceding vertices in the
sequence, and, at the end all vertices of $G$ are totally dominated (by
definition a vertex totally dominates its neighbors). The maximum length of a
total dominating sequence is called the Grundy total domination number,
$\gamma_{\rm gr}^t(G)$, of $G$, as introduced in [B. Bre\v{s}ar, M.A. Henning,
and D. F. Rall, Total dominating sequences in graphs, Discrete Math. 339
(2016), 1165--1676]. In this paper we continue the investigation of this
concept, mainly from the algorithmic point of view. While it was known that the
decision version of the problem is NP-complete in bipartite graphs, we show
that this is also true if we restrict to split graphs. A linear time algorithm
for determining the Grundy total domination number of an arbitrary tree $T$ is
presented, based on the formula $\gamma_{\rm gr}^t(T)=2\tau(T)$, where
$\tau(T)$ is the vertex cover number of $T$. A similar efficient algorithm is
presented for bipartite distance-hereditary graphs. Using the modular
decomposition of a graph, we present a frame for obtaining polynomial
algorithms for this problem in classes of graphs having relatively simple
modular subgraphs. In particular, a linear algorithm for determining the Grundy
total domination number of $P_4$-tidy graphs is presented. In addition, we
prove a realization result by exhibiting a family of graphs $G_k$ such that
$\gamma_{\rm gr}^t(G_k)=k$, for any $k\in{\mathbb{Z}^+}\setminus\{1,3\}$, and
showing that there are no graphs $G$ with $\gamma_{\rm gr}^t(G)\in \{1,3\}$. We
also present such a family, which has minimum possible order and size among all
graphs with Grundy total domination number equal to $k$.
