Consider a graph $G = (V, E)$ and a function $f: V \rightarrow \{0, 1, 2\}$. A vertex $u$ with $f(u)=0$ is defined as \emph{undefended} by $f$ if it lacks adjacency to any vertex with a positive $f$-value. The function $f$ is said to be a \emph{Weak Roman Dominating function} (WRD function) if, for every vertex $u$ with $f(u) = 0$, there exists a neighbour $v$ of $u$ with $f(v) > 0$ and a new function $f': V \rightarrow \{0, 1, 2\}$ defined in the following way: $f'(u) = 1$, $f'(v) = f(v) - 1$, and $f'(w) = f(w)$, for all vertices $w$ in $V\setminus\{u,v\}$; so that no vertices are undefended by $f'$. The total weight of $f$ is equal to $\sum_{v\in V} f(v)$, and is denoted as $w(f)$. The \emph{Weak Roman Domination Number} denoted by $\gamma_r(G)$, represents $min\{w(f)~\vert~f$ is a WRD function of $G\}$. For a given graph $G$, the problem of finding a WRD function of weight $\gamma_r(G)$ is defined as the \emph{Minimum Weak Roman domination problem}. The problem is already known to be NP-hard for bipartite and chordal graphs. In this paper, we further study the algorithmic complexity of the problem. We prove the NP-hardness of the problem for star convex bipartite graphs and comb convex bipartite graphs, which are subclasses of bipartite graphs. In addition, we show that for the bounded degree star convex bipartite graphs, the problem is efficiently solvable. We also prove the NP-hardness of the problem for split graphs, a subclass of chordal graphs. On the positive side, we give polynomial-time algorithms to solve the problem for $P_4$-sparse graphs. Further, we have presented some approximation results.

翻译：考虑图 $G = (V, E)$ 及函数 $f: V \rightarrow \{0, 1, 2\}$。若顶点 $u$ 满足 $f(u)=0$ 且不与任何 $f$ 值为正的顶点相邻，则称 $u$ 被 $f$ \emph{无防御}。函数 $f$ 被称为\emph{弱罗马控制函数}（WRD 函数），如果对于每个满足 $f(u) = 0$ 的顶点 $u$，都存在 $u$ 的一个邻点 $v$ 满足 $f(v) > 0$，并且可以按如下方式定义一个新函数 $f': V \rightarrow \{0, 1, 2\}$：$f'(u) = 1$，$f'(v) = f(v) - 1$，且对于 $V\setminus\{u,v\}$ 中的所有顶点 $w$，$f'(w) = f(w)$；使得没有顶点被 $f'$ 无防御。函数 $f$ 的总权重等于 $\sum_{v\in V} f(v)$，记作 $w(f)$。\emph{弱罗马控制数}记为 $\gamma_r(G)$，表示 $min\{w(f)~\vert~f$ 是 $G$ 的一个 WRD 函数$\}$。对于给定图 $G$，寻找权重为 $\gamma_r(G)$ 的 WRD 函数的问题被定义为\emph{最小弱罗马控制问题}。已知该问题对于二分图和弦图是 NP-难的。本文中，我们进一步研究了该问题的算法复杂性。我们证明了该问题对于星凸二分图和梳凸二分图（二分图的子类）是 NP-难的。此外，我们证明了对于有界度星凸二分图，该问题可以有效求解。我们还证明了该问题对于弦图的子类——分裂图是 NP-难的。在积极方面，我们给出了求解 $P_4$-稀疏图上该问题的多项式时间算法。此外，我们提出了一些近似结果。