The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility is given whenever (i) vertices labeled $\pm{}1$ have at least one neighbor with label in $\{2,3\}$; (ii) each vertex labeled $-1$ has one $3$-labeled neighbor or at least two $2$-labeled neighbors; and (iii) the sum of labels over the closed neighborhood of any vertex is positive. The cumulative weight of an optimal labeling is called signed double Roman domination number (SDRDN). In this work, we first consider the problem on general cubic graphs of order $n$ for which we present a sharp $n/2+\Theta(1)$ lower bound for the SDRDN by means of the discharging method. Moreover, we derive a new best upper bound. Observing that we are often able to minimize the SDRDN over the class of cubic graphs of a fixed order, we then study in this context generalized Petersen graphs for independent interest, for which we propose a constraint programming guided proof. We then use these insights to determine the SDRDNs of subcubic $2\times m$ grid graphs, among other results.

翻译：带符号双重罗马支配问题是一个组合优化问题，要求对图中每个顶点从集合$\{\pm{}1,2,3\}$中分配一个标签，使得所有标签的总和最小化，同时满足可行性条件：（i）标签为$\pm{}1$的顶点至少有一个邻居的标签属于$\{2,3\}$；（ii）每个标签为$-1$的顶点必须有一个标签为$3$的邻居，或至少两个标签为$2$的邻居；（iii）任意顶点闭邻域内的标签总和为正。最优标签方案的累积权重称为带符号双重罗马支配数（SDRDN）。本文首先考虑一般$n$阶立方图上的该问题，通过放电法证明了SDRDN的紧致下界为$n/2+\Theta(1)$，并推导出新的最优上界。注意到我们通常能在固定阶数的立方图类中最小化SDRDN，因此进一步独立研究了广义彼得森图，并提出了约束规划引导的证明方法。基于这些见解，我们还确定了次立方$2\times m$网格图的SDRDN，以及其他相关结果。