### On the total signed domination number of the Cartesian product of paths

#### Abstract

Let $G$ be a finite connected simple graph with a vertex set $V(G)$ and an edge set $E(G)$. A total signed dominating function of $G$ is a function $f: V(G)\cup E(G)\rightarrow \{-1, 1\}$, such that $\sum_{y\in N_T[x]}f(y) \geq 1$ for all $x\in V(G) \cup E(G)$. The total signed domination number of $G$ is the minimum weight of a total signed dominating function on $G$. In this paper, we prove lower and upper bounds on the total signed domination number of the Cartesian product of two paths, $P_m\Box P_n$.

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