*Submitted Paper*

**Inserted:** 5 apr 2016

**Last Updated:** 5 apr 2016

**Year:** 2016

**Abstract:**

We prove that there are no networks homeomorphic to the Greek ``theta'' letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self-similarly shrinking. This fact completes the classification of the self-similarly shrinking networks in the plane with at most two triple junctions.

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