On the Signed (Total) $k$-Domination Number of a Graph

Let $k$ be a positive integer and $G=(V,E)$ be a graph of minimum degree at least $k-1$. A function $f:V\rightarrow \{-1,1\}$ is called a \emph{signed $k$-dominating function} of $G$ if $\sum_{u\in N_G[v]}f(u)\geq k$ for all $v\in V$. The \emph{signed $k$-domination number} of $G$ is the minimum value of $\sum_{v\in V}f(v)$ taken over all signed $k$-dominating functions of $G$. The \emph{signed total $k$-dominating function} and \emph{signed total $k$-domination number} of $G$ can be similarly defined by changing the closed neighborhood $N_G[v]$ to the open neighborhood $N_G(v)$ in the definition. The \emph{upper signed $k$-domination number} is the maximum value of $\sum_{v\in V}f(v)$ taken over all \emph{minimal} signed $k$-dominating functions of $G$. In this paper, we study these graph parameters from both algorithmic complexity and graph-theoretic perspectives. We prove that for every fixed $k\geq 1$, the problems of computing these three parameters are all \NP-hard. We also present sharp lower bounds on the signed $k$-domination number and signed total $k$-domination number for general graphs in terms of their minimum and maximum degrees, generalizing several known results about signed domination.