Given a graph $G = (V, E)$, a signed dominating function is a function $f: V \rightarrow \{-1, 1\}$ such that for every vertex $u \in V$, $\sum\limits_{v \in N[u]} f(v) \geq 1$. The weight of $f$ is defined as $\sum\limits_{u \in V} f(u)$. The objective of the \sd{} problem is to compute a signed dominating function $f$ of minimum weight. The problem is known to be NP-complete even when restricted to bipartite, chordal, and planar graphs. In this paper, we extend the known complexity results for the \sd{} problem. Since the problem is NP-complete on chordal graphs, we study its complexity on split graphs, a subclass of chordal graphs, and show that it remains NP-complete. Moreover, as the problem is W[2]-hard parameterized by weight, we investigate its parameterized complexity with respect to structural parameters. We prove that the problem is W[1]-hard when parameterized by feedback vertex set number (and hence by treewidth and clique-width). Motivated by this hardness result, we consider more restrictive parameters, neighbourhood diversity and twin cover number, and present FPT algorithms.

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