Signed Total Roman Domination and Domatic Numbers: Degree Three and Complete Multipartite Graphs

Signed total Roman domination is a variant of the classic Roman domination-problem in graphs. A signed total Roman dominating function (STRD function) on a graph $G=(V,E)$ is a function $f: V \to \{-1,1,2\}$ such that (i) $\sum_{u \in N(v)} f(u) \geq 1$ for all $v \in V$, where $N(v)$ denotes the neighborhood of $v$, and (ii) every vertex $v$ with $f(v) = -1$ is adjacent to a vertex $u$ with $f(u) = 2$. The weight of $f$ is $\sum_{v \in V} f(v)$. The signed total Roman domination number of $G$ is the minimum weight among all its STRD functions. A signed total Roman dominating family (STRD family) on $G$ is a family $\{f_1, \ldots, f_d\}$ of pairwise distinct STRD functions such that $\sum_{i=1}^{d} f_i(v) \leq 1$ for all $v \in V$. The signed total Roman domatic number of $G$ is the maximum size among all its STRD families.
In this paper, we relate the signed total Roman domination number of a cubic graph to its open packing number, 2-tuple total domination number, and signed total domination number, allowing us to derive sharp bounds on the first invariant and to establish new $\mathcal{NP}$-completeness results for all four invariants. We demonstrate that having a degree-3 vertex determines a graph's signed total Roman domatic number. Combined with known results this implies that the associated decision problem is easy for graphs with maximum degree at most three and $\mathcal{NP}$-complete otherwise. To contrast these general hardness results, we determine signed total Roman domination and domatic numbers in complete multipartite graphs. Despite their simple structure, this is a non-trivial task.