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Probably the easiest way to thin and smooth polygons with a shared border is to:

1. Import your dataset of polygons.

2. Select the polygons and go to Tools | Thin and Smooth.

3. Select smoothing method and settings you want and click OK.

4. The shared lines will now be smoothed differently, possibly leaving gaps in between the lines.

5. Of the 2 polygons that have a shared border, select the polygon with the least desirable shared border.

6. Go to Tools | Change Boundary Type | Polygon to Polyline. That particular polygon is now a polyline.

7. Make sure that polyline is selected and go to Tool | Edit Boundaries | Trim Polyline and trim the polyline to remove the shared line segment. You just want the polyline without the shared line.

8. Once the shared portion of the polyline is removed, select the polyline and the other polygon and go to Tools | Edit Boundaries | Polyline to Polygon with Shared Border. Now the polyline is converted back to a polygon and the shared border is exact.