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The main functional purpose of a water distribution network is to transport water from a source to several domestic and industrial units while at the same time satisfying various requirements on hydraulic response. All the water distribution networks perform two basic operations: firstly the water network needs to deliver adequate amounts of water to meet specific requirements, and secondly the water network needs to be reliable therefore, the required amount of water needs to be continuously available 24 hours a day and 365 days per year. Due to the inevitable failures of some components such as pump stations, reservoirs and/or pipelines in a large-scale water distribution network, in designing a reliable network, the topological structure with low vulnerability must be achieved. Consequently, the study of connectivity, which is the key graph-theoretical notion, becomes crucial. This paper highlights some fundamental concepts from graph theory for vulnerability assessment of water distribution networks, addresses the mathematical properties of the link and node-deletion problems, and outlines some well-established results on the deterministic measures to assess the fault tolerance of networks.

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