Theoretical modelling of efficient fire safety water networks by certified domination

This paper explores a new way of designing water supply networks for fire safety using ideas from graph theory, focusing on a method called certified domination. Ensuring a good water supply is crucial for fire safety in communities, this study looks at the rules and problems in Poland for how much water is needed to fight fires in different areas and how this can be achieved at a lowest possible cost. We present a way to plan water supply networks for fire protection as a graph, where each point (node) is a place that needs water, and the lines (links) show where water can go between these points. The main idea is to find the best places to put pumping stations and wells in the network to save money and still meet all the fire safety requirements. Our approach assumes that it costs more to build a pumping station than a well. We use some examples to show how this method can find cost-effective solutions for water supply networks, while ensuring that they meet fire safety requirements and are not too expensive to build. This approach is a new and efficient way to improve the design of water supply networks for fire safety. Key challenges that are solved in this paper are a linear time algorithm finding an optimal solution for networks without cycles and a BLP (Binary Linear Programming) algorithm solving the problem in arbitrary networks.