AI 中文总结
研究凸二分图和弦二分图上四种罗马支配变体，为凸二分图开发统一动态规划框架，可\(O(n^6)\)时间计算相关参数，同时证明弦二分图上部分变体仍NP完全，明确两类图算法差异。
AI 中文摘要
罗马支配及其变体是由保护、容错和资源分配驱动的一类重要的支配型图参数。图\(G\)的罗马支配函数是一个函数\(f:V(G)\rightarrow\{0,1,2\}\)，使得每个\(f(v)=0\)的顶点\(v\)都有一个\(f(u)=2\)的邻居\(u\)。\(f\)的权重是\(w(f)=\sum_{v\in V(G)}f(v)\)，\(G\)的罗马支配函数的最小权重是罗马支配数，记为\(\gamma_R(G)\)。本文研究二分图的两个自然子类，即凸二分图和弦二分图上的四种罗马支配变体。一方面，为凸二分图上的罗马-\(\{2\}\)支配、双罗马支配、完美罗马支配和唯一响应罗马支配开发了一个统一的从左到右的动态规划框架。算法利用一个二分划类的区间结构，并使用常数数量的边界索引来表示所有未完成的需求。因此，这四个参数中的每一个都可以在\(O(n^6)\)时间内计算出来，其中\(n = |V(G)|\)。另一方面，证明了罗马-\(\{2\}\)支配、完美罗马支配和唯一响应罗马支配在弦二分图上仍然是NP完全的。这些结果在凸二分图（区间排序产生多项式时间可解性）和更广泛的弦二分图类（几个罗马型支配问题仍然计算上难以处理）之间建立了明确的算法分离。
英文摘要
Roman domination and its variants form an important family of domination-type graph parameters motivated by protection, fault tolerance, and resource allocation. A Roman dominating function of a graph \(G\) is a function \(f:V(G)\rightarrow\{0,1,2\}\) such that every vertex \(v\) with \(f(v)=0\) has a neighbour \(u\) with \(f(u)=2\). The weight of \(f\) is \(w(f)=\sum_{v\in V(G)}f(v)\), and the minimum weight of a Roman dominating function of \(G\) is the Roman domination number, denoted by \(γ_R(G)\). In this paper, we study four variants of Roman domination on two natural subclasses of bipartite graphs, namely convex bipartite graphs and chordal bipartite graphs. On the positive side, we develop a unified left-to-right dynamic programming framework for Roman-\(\{2\}\) domination, double Roman domination, perfect Roman domination, and unique response Roman domination on convex bipartite graphs. The algorithms exploit the interval structure of one bipartition class and represent all unfinished requirements using a constant number of boundary indices. Consequently, each of the four parameters can be computed in \(O(n^6)\) time, where \(n=|V(G)|\). On the negative side, we prove that Roman-\(\{2\}\) domination, perfect Roman domination, and unique response Roman domination remain NP-complete on chordal bipartite graphs. These results establish a clear algorithmic separation between convex bipartite graphs, where the interval ordering yields polynomial-time solvability, and the broader class of chordal bipartite graphs, where several Roman-type domination problems remain computationally intractable.
