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Link failures in supply networks can have catastrophic consequences that canlead to a complete collapse of the network. Strategies to prevent failurespreading are thus heavily sought after. Here, we make use of a spanning treeformulation of link failures in linear flow networks to analyse topologicalstructures that prevent failures spreading. In particular, we exploit a resultobtained for resistor networks based on the \textit{Matrix tree theorem} toanalyse failure spreading after link failures in power grids. Using a spanningtree formulation of link failures, we analyse three strategies based on thenetwork topology that allow to reduce the impact of single link failures. Allour strategies do not reduce the grid's ability to transport flow or do in factimprove it - in contrast to traditional containment strategies based onlowering network connectivity. Our results also explain why certainconnectivity features completely suppress any failure spreading as reported inrecent publications.

Topological theory of resilience and failure spreading in flow networks