are identified backwards starting from panel 4 to 1. Panel 6 shows the two sources identified separately, with the [ *bridge pixels*]{} re-assigned according to a local criterion.[]{data-label="separate"}](separate.ps){width="\linewidth"}

There are some cases that require special treatments. The first one concerns blending of extended sources. In principle, blended extended sources cannot be separated without making assumptions on their surface brightness distributions. However, for most of the extended sources we are interested in, such as regular clusters with only one bright cool core, it is reasonable to assume that the surface brightness decreases monotonically from the core to the edge. In such cases, blended extended sources can be separated with the same method as described above. This step is left as an option. The second case is the bridging between extended sources and background overdensity regions, which results in branch-like shapes at the edge of extended sources. We managed to clean such cases by identifying the [*cut-vertex*]{}. A [*cut-vertex*]{} is similar to a [*bridge pixel*]{}, but it is defined as the single pixel whose removal allows one to separate the blended sources into two or more components in the image. Clearly, the presence of a [*cut-vertex*]{} indicates a weak connection between two 