  A vertex $v$ of a graph $G=(V,E)$ is said to be undefended with respect to a
function $f: V \longrightarrow \{0,1,2\}$ if $f(v)=0$ and $f(u)=0$ for every
vertex $u$ adjacent to $v$. We call the function $f$ a weak Roman dominating
function if for every $v$ such that $f(v)=0$ there exists a vertex $u$ adjacent
to $v$ such that $f(u)\in \{1,2\}$ and the function $f': V \longrightarrow
\{0,1,2\}$ defined by $f'(v)=1$, $f'(u)=f(u)-1$ and $f'(z)=f(z)$ for every
$z\in V \setminus\{u,v\}$, has no undefended vertices. The weight of $f$ is
$w(f)=\sum_{v\in V(G) }f(v)$. The weak Roman domination number of a graph $G$,
denoted by $\gamma_r(G)$, is the minimum weight among all weak Roman dominating
functions on $G$. Henning and Hedetniemi [Discrete Math. 266 (2003) 239-251]
showed that the problem of computing $\gamma_r(G)$ is NP-Hard, even when
restricted to bipartite or chordal graphs. This suggests finding $\gamma_r(G)$
for special classes of graphs or obtaining good bounds on this invariant. In
this article, we obtain closed formulae and tight bounds for the weak Roman
domination number of lexicographic product graphs in terms of invariants of the
factor graphs involved in the product.
