AI 中文总结
本文研究可通过连接、并操作分解为两子图的图的{2}-罗马性质，刻画了Partner Limited图相关特定图族的{2}-罗马图分类，推进了{2}-罗马图识别问题的研究。
AI 中文摘要
给定图$G=(V,E)$，若函数$f: V \rightarrow \{0,1,2\}$满足：对每个满足$f(v)=0$的顶点$v \in V$，要么存在与$v$相邻的顶点$u$使得$f(u)=2$，要么存在两个与$v$相邻的不同顶点$x,y$使得$f(x)=f(y)=1$，则称$f$为$G$的罗马{2}-控制函数（Chellali等人，2016）。对于任意图$G$，都满足$\gamma_{\{R2\}}(G) \leq 2\gamma(G)$，其中$\gamma_{\{R2\}}(G)$表示$G$的{2}-罗马控制函数的最小权重，$\gamma(G)$为$G$的控制数。{2}-罗马图是指满足上述等式的图（Klostermeyer等人，2019）。Henning等人于2017年给出了{2}-罗马树的刻画；2025年，Ferrari等人通过$G$仅取0、2值的最小{2}-罗马控制函数的存在性，刻画了{2}-罗马性质；同年，Bešter Štorgel等人提出了{2}-罗马图的识别问题，证明了中间图的识别问题具有多项式复杂度，并刻画了可通过分裂连接操作分解为两个较小分裂图的{2}-罗马分裂图，而一般图的识别复杂度仍未解决。本文研究可通过连接、并操作分解为两个较小图的{2}-罗马性质，实现了{2}-罗马性质的完整刻画；4-路径是顶点和边数最少的非平凡连通非{2}-罗马图。我们在Partner Limited图分解中出现的、含有限个4-路径的特定不可分解图族中，对{2}-罗马性质进行了分类，这些图包括良标记蜘蛛、ZOO中的图及一些特殊分裂图。
英文摘要
Given a graph $G=(V,E)$, $f : V \rightarrow \{0, 1, 2\}$ is a \emph{Roman $\{2\}$-dominating function} of $G$ if for every vertex $v\in V$ with $f(v) =0$, either there exists a vertex $u$ adjacent to $v$ with $f(u) = 2$, or two distinct vertices $x,\; y$ both adjacent to $v$ with $f(x)=f(y)=1$ (Chellali et al. 2016). Every graph $G$ satisfies $γ_{\{R2\}}(G) \leq 2γ(G)$, where $γ_{\{R2\}}(G)$ denotes the minimum weight of a $\{2\}$-Roman dominating function of $G$ and $γ(G)$ is the domination number of $G$. \emph{$\{2\}$-Roman graphs} are those for which the equality is reached (Klostermeyer et al. 2019). A characterization of $\{2\}$-Roman trees was given by Henning et al. in 2017. In 2025, Ferrari et al. characterized the $\{2\}$-Roman property by the existence of a minimum $\{2\}$-Roman dominating function of $G$ that assumes only $0, 2$-values. Afterwards in 2025, Bešter Štorgel et al. introduced the problem of recognizing $\{2\}$-Roman graphs, proved polinomiality for middle graphs, and characterized \hbox{$\{2\}$-Roman} split graphs that can be decomposed with respect to the split join operation into two smaller split graphs. Recognition complexity is still open for general graphs. In this paper we study the \hbox{$\{2\}$-Roman} property on graphs that can be decomposed into two smaller graphs with respect to the join and union operations, allowing to completely characterize the $\{2\}$-Roman property. The 4-path is the non trivial connected non $\{2\}$-Roman graph with the fewest number of vertices and edges. We classify the $\{2\}$-Roman property within specific families of non decomposable graphs with a limited number of 4-paths which are present in the decomposition of partner limited graphs; these are well-labelled spiders, the graphs in ZOO and some special split graphs.
