Simplify polygons without creating slithers

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When simplifying polygons it’s almost inevitable that you will generate some slither polygons or gaps between the correct polygons. For example, the following image shows two adjoining complex polygons, representing two adjoining administrative areas. Note there are no gaps between the polygons; they are contiguous (the border is between Sheffield and Barnsley LADs, by the way).Original geometries without simplification Now if we simplify these polygons to reduce their complexity using simplify geometries in QGIS, gaps appear between the original two polygons and they are no longer contiguous.