# On the weak Roman domination number of lexicographic product graphs

## TL;DR

This paper derives formulas and bounds for the weak Roman domination number of lexicographic product graphs, a complex graph invariant, in terms of the properties of the component graphs, addressing computational hardness issues.

## Contribution

It provides the first closed-form formulas and tight bounds for the weak Roman domination number of lexicographic product graphs.

## Key findings

- Derived closed formulas for the weak Roman domination number.
- Established tight bounds based on factor graph invariants.
- Addressed computational complexity by focusing on special graph classes.

## Abstract

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.