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In QGIS the processing algorithm 'Validate Coverage' identifies cases of non exactly matching edges between adjacent polygons. Like in the picture below : although vertices 1. 2. and 3. of the left polygon are aligned (or quasi-aligned due to float64 rounding), only vertices 1. and 3. belong to the right polygon and so the edges don't match, creating a flat triangular gap and slicing the common boundary in two parts.

Is there an algorithm that takes the output of 'Validate Coverage' and inserts the missing vertex in the right polygon?

Incorrect vertex coverage.

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    You can try the Snap Geometries to Layer tool. My preference is to use the v.clean tool in the GRASS section using the break, snap and bpol tools. There is a rmdupl but I prefer to run the delete duplicate geometries afterwards.
    – John
    Commented Nov 2 at 17:36

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I made tests with QGIS and with OpenJUMP GIS and I think both of them can fix the topology of your data.

As a test data I used two simple adjacent polygons, each one having one vertex that was missing from the other polygon.

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I tried the QGIS Snap Geometries to Layer tool with these settings.

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The tool made the polygon boundaries to match by adding one vertex to the polygon on the right, and by removing one vertex.

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OpenJUMP GIS has a tool Coverage Cleaner (that still shows the old name Adjust Polygon Boundaries in the title).

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Unlike QGIS, this tool does not remove any vertex, even it could do that without breaking the topology. OpenJUMP adds the missing vertices from each polygon to the other.

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