We show that in almost every random graph process, the hitting time for having k edge-disjoint spanning trees equals the hitting time for having minimum degree k.

Background Stakeholder and community engagement promotes collaboration and gives service users an opportunity to actively participate in the care they receive. Recognizing this potential, The Network for Improving Quality of Care for maternal, newborn and child health aimed to identify tools and operational guidance to integrate stakeholder and community engagement into quality improvement (QI) implementation. Methods A mapping, consisting of a literature review and an open call through email and listservers, for implementation tools was conducted. Materials were included if they provided guidance on stakeholder and community engagement aligned to the Network's QI framework comprising seven phases. Screening of tools was done by two reviewers. Results The literature search and the call for tools returned 197 documents with 70 tools included after screening. Most included tools (70%) were published after 2010. International organizations were the most frequently cited authors of tools. Only 15 tools covered all seven phases of the QI framework; few tools covered the more ‘technical’ phase of the QI framework: adapting standards and refining strategies. Conclusion The quantity of tools and their varied characteristics including types of stakeholder and community engagement processes across the QI framework confirms that engagement cannot be captured in a ‘one‐size‐fits‐all’ formula. Many tools were designed with a generic focus to allow for adaption and use in different settings and sectors. Country programmes looking to strengthen engagement approaches can take advantage of available tools through an online portal on the WHO website and adapt them to meet their specific needs and context. Public Involvement Programme implementers provided tools and resources during data collection.

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