Total dominating sequences in trees, split graphs, and under modular decomposition

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 $k$.