基于彩虹与罗马控制参数的灾后避难营地空间建模的图论方法

中文摘要 AI 辅助

灾后避难营地的有效空间组织对于确保基本服务的可及性以及高效利用有限的空间和资源至关重要。本文提出了一种基于彩虹 $k$-控制（rainbow $k$-domination）和罗马控制（Roman domination）的图论框架，用于避难营地设施的布局。彩虹 $k$-控制模型模拟了不同类型设施（如卫生单元、厨房、供水点和学校）的同时可及性，而罗马控制则用于表示具有不同容量水平的服务，通过Wi-Fi部署加以说明。我们提出了一种 $O(nk)$ 时间算法，用于在具有 $n$ 个顶点的树中找到最小彩虹 $k$-控制集，对于固定的 $k$，该算法在 $n$ 上是线性的，计算实验证实了其在大规模实例上的可扩展性。对于一般图，我们建立了彩虹 $k$-控制数的新下界和上界。我们进一步研究了其与罗马控制的关系，推导了达到两个参数之间极值相等性的图的结构性质，并证明了识别此类图是NP难的。这些结果为基于控制的灾后避难规划中设施布局方法提供了理论基础。

英文摘要

Effective spatial organization of post-disaster shelter camps is essential for ensuring access to basic services while making efficient use of limited space and resources. In this paper, we propose a graph-theoretic framework for shelter-camp facility placement based on rainbow $k$-domination and Roman domination. Rainbow $k$-domination models the simultaneous accessibility of distinct facility types, such as sanitation units, kitchens, water points, and schools, whereas Roman domination is used to represent services with different capacity levels, illustrated through Wi-Fi deployment. We present an $O(nk)$-time algorithm for finding a minimum rainbow $k$-dominating set of a tree with $n$ vertices, which is linear in $n$ for fixed $k$, and computational experiments confirm its scalability on large instances. For general graphs, we establish new lower and upper bounds on the rainbow $k$-domination number. We further investigate its relationship with Roman domination, derive structural properties of graphs attaining the extremal equality between the two parameters, and prove that recognizing such graphs is NP-hard. These results provide a theoretical foundation for domination-based approaches to facility placement in post-disaster shelter planning.

发表机构

• Gebze Technical University（盖布泽理工大学）
• Gdańsk University of Technology（格但斯克理工大学）

机构由 AI 辅助整理，请以论文原文为准。